# Membrane-Mediated Interactions Between Nonspherical Elastic Particles

Jiarul Midya<EMAIL_ADDRESS>Thorsten Auth<EMAIL_ADDRESS>Gerhard Gompper  Theoretical Physics of Living Matter, Institute for
Biological Information Processing and Institute for Advanced Simulation,
j.midya@ Jülich, 52425 Jülich, Germany<EMAIL_ADDRESS>
###### Abstract

The transport of particles across lipid-bilayer membranes is important for
biological cells to exchange information and material with their environment.
Large particles often get wrapped by membranes, a process which has been
intensively investigated in the case of hard particles. However, many
particles in vivo and in vitro are deformable, e.g., vesicles, filamentous
viruses, macromolecular condensates, polymer-grafted nanoparticles, and
microgels. Vesicles may serve as a generic model system for deformable
particles. Here, we study non-spherical vesicles with various sizes, shapes,
and elastic properties at initially planar lipid-bilayer membranes. Using the
Helfrich Hamiltonian, triangulated membranes, and energy minimization, we
predict the interplay of vesicle shapes and wrapping states. Increasing
particle softness enhances the stability of shallow-wrapped and deep-wrapped
states over non-wrapped and complete-wrapped states. The free membrane
mediates an interaction between partial-wrapped vesicles. For the pair
interaction between deep-wrapped vesicles, we predict repulsion. For shallow-
wrapped vesicles, we predict attraction for tip-to-tip orientation and
repulsion for side-by-side orientation. Our predictions may guide the design
and fabrication of deformable particles for efficient use in medical
applications, such as targeted drug delivery.

###### keywords:

particle uptake, passive endocytosis, particle deformability, prolate
vesicles, wrapping states, membrane-mediated interactions, continuum membrane
model

Wrapping processes of mesoscopic particles by cell membranes are abundant in
nature, such as entry and exit of viruses and parasites into host cells 1, 2.
Furthermore, extracellular vesicles can be taken up via endocytosis 3.
Therefore, particle wrapping by lipid-bilayer membranes is also important for
the design diagnostic and therapeutic agents. Particles for medical
applications include lipid particles for targeted drug delivery 4, 5, and
magnetic nanoparticles that serve as heat sources for cancer therapy 6. For
hard particles the desired wrapping states can be achieved by controlling
their shape, size, and particle-membrane adhesion strength 7. For soft
particles, in addition, particle deformability plays an important role in the
wrapping process. For example, an increased stability of partial-wrapped
states has been reported for initially spherical vesicles with low bending
rigidities of their membrane 8. Vesicles flatten upon binding to planar
substrates 9, the increased stability of partial-wrapped states is therefore
similar to the increased stability reported for hard oblate ellipsoidal
compared with spherical particles 10. Furthermore, soft particles can adjust
to constraints and react to external stimuli. Prominent examples are
filamentous viruses at plasma membranes that bend 11, parasites that squeeze
through narrow constrictions when invading host cells through the tight
junction 12, and SARS-CoV-2 virions, which are bounded by a lipid envelope,
and assume non-spherical shapes near membranes 13.

Engineered soft particles encompass a wide variety of architectures with
tunable elastic properties: microgels 14, star polymers 15, polymer-grafted
nanoparticles 16, polymeric shells 17, particles made from dendrimers as
building blocks 18, vesicles 19, and biomolecular condensates 20. For example,
the elastic properties of microgels can be controlled by crosslinker density
and electric charge 21, 22, of star polymers by functionality and chain length
23, of polymer-grafted nanoparticles by grafting density and chain length 16,
and of unilamellar fluid vesicles by membrane bending rigidity and osmotic
concentrations 24. Vesicles, in particular, are a versatile and well-
established biomimetic system and have also been used as generic model system
for soft particles 25. A zoo of vesicle shapes can be obtained by changing the
membrane curvature-elastic parameters and osmotic concentrations, which
includes spherical, prolate, oblate, stomatocyte, pear-shaped, and starfish
shapes 19.

Particles have been shown to laterally move on lipid membranes in experiments
using viruses on plasma membranes and synthetic particles on free-standing
lipid bilayers, as well as in computer simulations 26, 27, 28. Therefore,
long-ranged membrane-mediated interactions can drive self-assembly of partial-
wrapped particles at lipid-bilayer membranes 29, 30. However, the nature of
the membrane-mediated interactions can either be attractive or repulsive,
depends on many parameters, and is thus not easy to predict. For example,
both, the deformation energy of the free membrane and the particle-membrane
adhesion energy depend on the distance between interacting particles 31.
Experimental and theoretical studies show membrane-mediated attraction and
cluster formation of hard spherical particles on vesicles 32. For example,
hard spherical particles on giant unilamellar vesicles have been found to
attract each other and to induce tube formation 33, 31, 34, 35, 36. In vivo,
viruses are shed as multi-virion clusters in vesicles 37, 38. Ellipsoidal
microgels on giant unilamellar vesicles have been found to form ordered
structures with spacings that suggest membrane-mediated mutual repulsion 14.

In this work, we use energy minimization to calculate and predict shapes and
wrapping states for single, non-spherical vesicles that get wrapped at planar
membranes. We find that adhesion to a membrane can change the shape of
vesicles from prolate to oblate for partial-wrapped states. Wrapping
transitions can be continuous and discontinuous; shape and orientation changes
of vesicles are always discontinuous with an energy barrier. Increased vesicle
softness, decreased reduced vesicle volume, and increased tension of the
planar membrane enhance the stability of the partial-wrapped states. We also
calculate the membrane-mediated pair interaction between two prolate vesicles
in shallow and deep-wrapped states. We predict attraction between shallow-
wrapped vesicles in tip-to-tip orientation, and repulsion between deep-wrapped
vesicles, as well as between shallow-wrapped vesicles in side-by-side
orientation. For shallow-wrapped vesicles, increasing particle deformability
induce a shape change from prolate to oblate and an interaction change from
attractive in tip-to-tip orientation to repulsive for oblate vesicles.

##  1 Results and Discussion

Wrapping energies. Our predictions for wrapping vesicles at membranes are
based on a continuum membrane model where the membranes are represented by
mathematical surfaces, which is applicable for particles with radii $r_{\rm
p}\gtrsim 20\,\rm nm$. The total energy of the system is calculated using the
energy functional

$E=\int_{A_{\rm p}}dS\,[2\kappa_{\rm p}H^{2}+\sigma]+\int_{A_{\rm
v}}dS\,2\kappa_{\rm v}H^{2}-w\int_{A_{\rm ad}}dS\,,$ (1)

where $H=(c_{1}+c_{2})/2$ is the mean curvature and $c_{1}$ and $c_{2}$ are
the two principle curvatures; $A_{\rm v}$ is the area of the vesicle membrane,
$A_{\rm p}$ the area of the initially planar membrane, and $A_{\rm ad}$ is the
adhered area of the vesicle membrane. The bending rigidities are $\kappa_{\rm
p}$ and $\kappa_{\rm v}$ for the planar and the vesicle membrane,
respectively; $\sigma$ is the tension of the planar membrane, and $w$ is the
adhesion strength between vesicle and membrane. The shape of the vesicle is
characterized by its reduced volume $v=V_{\rm v}/V_{\rm sph}$, where $V_{\rm
v}$ is the actual volume of the vesicle and $V_{\rm sph}$ is the volume of a
spherical vesicle with the same membrane area. The deformability of the
vesicle is characterized by the ratio $\kappa_{\rm v}/\kappa_{\rm p}$ of the
bending rigidities of vesicle and planar membrane. The wrapping fraction
$f_{w}=A_{\rm ad}/A_{\rm v}$ measures the fraction of the vesicle membrane
that is adhered. For convenience, we will use the reduced membrane tension
$\tilde{\sigma}=\sigma A_{\rm v}/(2\pi\kappa_{\rm p})$ and the reduced
adhesion strength $\tilde{w}=wA_{\rm v}/(\pi\kappa_{\rm p})$ in the following.

Wrapping and shape transitions for a non-spherical vesicle. The wrapping of
non-spherical vesicles at a planar membrane is studied by systematically
varying the bending rigidity ratio $\kappa_{\rm v}/\kappa_{\rm p}$ and reduced
volume $v$ of the vesicle, and the tension $\sigma$ of the planar membrane.
Figure 1 shows minimum energy shapes of a vesicle with reduced volume $v=0.8$,
at a planar membrane as function of the wrapping fraction $f_{w}=A_{\rm
ad}/A_{\rm v}$. Wrapping always starts from the lowest mean curvature region
of the vesicle surface; for a prolate vesicle binding occurs in submarine
orientation with its major axis parallel to the membrane, see Fig. 1 (a). As
wrapping progresses, the deformation of vesicle and surrounding membrane
increases, see Fig. 1(b)-(d). We observe a stable oblate state of the vesicle
at $f_{w}\approx 0.3$. The deformation of the vesicles reaches its maximum at
$f_{w}\approx 0.5$, where the vesicle shape is oblate and the membrane touches
the rim of the vesicle where the mean curvature is maximal. For $f_{w}>0.5$,
the shape of the vesicle changes back from oblate to prolate–now with rocket
orientation and its major axis perpendicular to the membrane, see Fig. 1(e,f).

Figure 1: Wrapping of a non-spherical vesicle with $v=0.8$ at an initially
planar membrane for the wrapping fractions (a) $f_{w}=0.05$ (unstable state),
(b) $f_{w}=0.25$ (stable state), (c) $f_{w}=0.35$ (stable state), (d)
$f_{w}=0.5$ (unstable state), (e) $f_{w}=0.75$ (stable state), and (f)
$f_{w}=0.95$ (stable state). The minimum-energy shape of the free vesicle is
prolate. The bending rigidity ratio is $\kappa_{\rm v}/\kappa_{\rm p}=1$ and
the reduced membrane tension $\tilde{\sigma}=0.5$. The left, middle, and right
columns represent the side, bottom, and top views of the vesicle-membrane
system, respectively.

In absence of adhesion energy ($\tilde{w}=0$), a monotonic increase of the
wrapping energy $\Delta\tilde{E}$ is observed with increasing wrapping
fraction $f_{w}$, see Fig. 2 (a). At finite adhesion strengths, depending on
the elastic properties, three or four transitions between non-wrapped,
partial-wrapped, and complete-wrapped states are found. For adhesion strengths
below the binding transition, $\tilde{w}<\tilde{w}_{1}$, the non-wrapped state
is stable. The binding transition $W_{1}$ at adhesion strength $w_{1}$ is
discontinuous, see Fig. 6S in the Supporting Information, corresponding to the
merging of two separate adhesion patches, see Fig. 1. At higher adhesion
strength, a transition from shallow-wrapped prolate in submarine orientation
to shallow-wrapped oblate may occur. A further increase of the adhesion
strength yields a coexistence of a shallow-wrapped state with low wrapping
fraction and a deep-wrapped state of a prolate in rocket orientation with high
wrapping fraction; this $W_{2}$ transition at adhesion strength $w_{2}$ is
also discontinuous. The envelopment transition $W_{3}$ at adhesion strength
$w_{3}$, between a deep-wrapped and the complete-wrapped state, is continuous.
For adhesion strengths above the envelopment transition,
$\tilde{w}\geq\tilde{w}_{3}$, the complete-wrapped state is stable.

Figure 2: Wrapping of a non-spherical vesicle at a planar membrane. (a)
Reduced wrapping energy $\Delta\tilde{E}$ as function of the wrapping fraction
$f_{w}$ for vesicles with reduced volume $v=0.8$, bending rigidity ratio
$\kappa_{\rm v}/\kappa_{\rm p}=5$, and reduced membrane tension
$\tilde{\sigma}=0.5$. The blue circles show the numerical data for the
wrapping energy in absence of adhesion strength ($\tilde{w}=0$) and the blue
line represents the fit function. The green, red, and orange lines correspond
to the adhesion strengths $\tilde{w}$ associated with the wrapping transitions
from non-wrapped (NW) to shallow-wrapped (SW), shallow-wrapped to deep-wrapped
(DW), and deep-wrapped to complete-wrapped (CW), respectively. The energies
for the (coexisting) stable states at the transitions are indicated by
circles. (b) Wrapping diagram in the $\kappa_{\rm v}/\kappa_{\rm
p}$-$\tilde{w}$ plane for $v=0.8$ and $\tilde{\sigma}=0.5$. The dashed red
line of the oblate-to-prolate transition is a guide to the eye. (c) Wrapping
diagram in the $v$-$\tilde{w}$ plane for $\kappa_{\rm v}/\kappa_{\rm p}=1$ and
$\tilde{\sigma}=0.5$. (d) Wrapping diagram in the $\tilde{\sigma}$-$\tilde{w}$
plane for $v=0.8$ and $\kappa_{\rm v}/\kappa_{\rm p}=1$. All transitions and
the stable wrapping states of the vesicles are labeled in the figures.

The wrapping diagrams of the vesicle-membrane system are obtained by
systematic variation of the bending-rigidity ratio $\kappa_{\rm v}/\kappa_{p}$
that effectively controls the vesicle softness, the reduced volume $v$ that
controls the shape of free vesicles, see Fig. 1S in Supporting Information,
and the reduced tension $\tilde{\sigma}$ of the planar membrane. For each
parameter set, we compute the deformation energy $\Delta\tilde{E}$ as function
of the wrapping fraction $f_{w}$, see Fig. 3S-5S in the Supporting
Information. Figure 2(b) shows the wrapping diagram in the
$\tilde{w}$-$\kappa_{\rm v}/\kappa_{p}$ plane for fixed $v=0.8$ and
$\tilde{\sigma}=0.5$. As we decrease the value of $\kappa_{\rm v}/\kappa_{p}$,
the binding transition $W_{1}$ occurs at lower adhesion strength $\tilde{w}$;
in contrast, the adhesion strength for the envelopment transition increases.
Therefore, as the vesicle becomes softer, binding to the planar membrane
becomes easier and complete wrapping more difficult. For the $W_{2}$
transition from a shallow-wrapped to a deep-wrapped state, a very weak
increase of $\tilde{w}$ is observed with decreasing $\kappa_{\rm
v}/\kappa_{p}$, i.e., the stability of the SW over the DW state increases.
Within the shallow-wrapped state, a vesicle shape transformation occurs
between soft oblate and stiff prolate vesicles.

Figure 3: Wrapping fractions $f_{w}$ for the shallow-wrapped states that
coexist with the non-wrapped state at the discontinuous binding transition
$W_{1}$, and for the coexisting shallow-wrapped and deep-wrapped states that
coexist at the $W_{2}$ transition. The wrapping fractions are presented as
function of (a) $\kappa_{\rm v}/\kappa_{\rm p}$ for fixed $v=0.8$ and
$\tilde{\sigma}=0.5$, (b) $v$ for fixed $\kappa_{\rm v}/\kappa_{\rm p}=1$ and
$\tilde{\sigma}=0.5$, and (c) $\tilde{\sigma}$ for fixed $v=0.8$ and
$\kappa_{\rm v}/\kappa_{\rm p}=1$. The dashed lines represent discontinuities
in $f_{w_{1}}$ associated with shape changes of the vesicles.

The effect of the tension $\tilde{\sigma}$ of the planar membrane on the
wrapping diagram is shown in Fig. 2(c). The adhesion strength $w_{1}$ for the
binding transition $W_{1}$ is independent of $\tilde{\sigma}$. The adhesion
strength $w_{2}$ for the shallow- to deep-wrapped transition $W_{2}$ increases
strongly with increasing $\tilde{\sigma}$. The adhesion strength $w_{3}$ for
the envelopment transition $W_{3}$ increases only weakly for small
$\tilde{\sigma}$ and strongly for high $\tilde{\sigma}$. Finally, the reduced
volume of the vesicles is varied in the range $0.7\leq v\leq 0.95$, where free
vesicles are prolate, see Fig. 2(d) and Fig. 1S in the Supporting Information.
The adhesion strengths for all three transitions increase with decreasing $v$,
accompanied by a change of the shape of the free vesicles that become more
elongated with their tips more pointed. Therefore, binding as well as complete
wrapping become more difficult, and the stability of the partial-wrapped
states increases.

Figure 3 shows the wrapping fractions for partial-wrapped states that coexist
with the free vesicles at the $W_{1}$ transition, and for shallow-wrapped and
deep-wrapped states that coexist at the $W_{2}$ transition. For the $W_{2}$
transition, the wrapping fractions $f_{w_{2}}^{\rm SW}$ for the SW states and
$f_{w_{2}}^{\rm DW}$ for the DW state slowly increase with decreasing
$\kappa_{\rm v}/\kappa_{\rm p}$ at reduced volume $v=0.8$ and tension
$\tilde{\sigma}=0.5$, see Fig. 3(a). The stronger increase of $f_{w_{2}}^{\rm
SW}$ indicates the vanishing of DW states in the limit $\kappa_{\rm
v}/\kappa_{\rm p}\rightarrow 0$. For the $W_{1}$ transition, a sudden increase
in wrapping fraction of the partial-wrapped states $f_{w_{1}}$ is observed at
$\kappa_{\rm v}/\kappa_{\rm p}\simeq 1$, which is associated with the shape
change of the vesicle from prolate to oblate.

The wrapping fractions $f_{w_{2}}^{\rm SW}$ and $f_{w_{2}}^{\rm DW}$ remain
almost unchanged when the reduced volume is varied in the range $0.7\leq v\leq
0.95$ at $\kappa_{\rm v}/\kappa_{\rm p}=1$ and $\tilde{\sigma}=0.5$, see Fig.
3(b). However, the variation of $f_{w_{1}}$ with $v$ is discontinuous. A
sudden drop of $f_{w_{1}}$ is observed as $v$ increases from $0.7$ to $0.75$.
This is associated with the shape change of the vesicle from oblate to
prolate. For $v\geq 0.75$, $f_{w_{1}}$ decreases monotonically with $v$, and
then again a sudden drop is observed from $v=0.85$ to $v=0.9$. Such a behavior
is expected as the shape of the free vesicle changes from peanut-like to
ellipsoid-like. For $0.75<v<0.85$, the shape of free vesicle is peanut-like
(see Fig. 1S), thus, initial attachment to membrane leads to two patches, see
Fig. 1(a). The vesicle becomes ellipsoid-like for $v\gtrsim 0.9$, thus,
initial binding of the vesicle to the planar membrane occurs at the middle
part and forms a single patch, see Fig. 2S in the Supporting Information. For
both transitions, a weak increase of the wrapping fractions of the coexisting
partial-wrapped states is observed with increasing membrane tension
$\tilde{\sigma}$ at reduced volume $v=0.8$ and $\kappa_{\rm v}/\kappa_{\rm
p}=1$, see Fig. 3(c).

For all partial-wrapped states, the vesicle deforms the free membrane, see
Fig. 4. For partial-wrapped vesicles in submarine orientation, the deviation
of the height field describing the free membrane from the plane is strongest
around the tips and weakest along the sides. The local bending energy of the
free membrane is finite at both, tips and sides, and shows lines of vanishing
bending energy separating tips and sides, see Fig. 4(c). The free-membrane
deformation at the sides can be approximated by calculations for wrapping
infinitely long cylindrical particles 39, 40, the deformation at the tips by
calculations for wrapping spherical particles 10, 41. However, the free-
membrane deformation along the sides rises the particle height above the
optimal value for catenoid formation around the tips. The finite mean
curvature along the sides and at the tips thus originate from principal
curvatures with opposite signs. Therefore, the mean curvature and also the
local bending energy vanish along lines where the principal curvatures have
equal magnitude. For the deep-wrapped vesicles, the free membrane deformation
and the bending energy distribution are cylindrical symmetric, see Fig. 4(d).
Here, the origin of the finite bending energy of the free membrane is the
membrane tension that prevents a catenoidal shape of the neck.

Figure 4: Single vesicles attached from above to initially planar membranes.
The heat maps indicate (a,b) the height and (c,d) the bending energy of the
free membrane around a vesicle with $v=0.8$, $\kappa_{\rm v}/\kappa_{\rm
p}=5$, $\tilde{\sigma}=0.5$ and $\tilde{w}=10.94$ for (a,b) shallow-wrapped
and (c,d) deep-wrapped states. The white areas are inside the contact line
where the free membrane detaches from the vesicle. Figure 5: Deep-wrapped
vesicles at $\tilde{w}=15$. (a) Interaction potential $\Delta\tilde{E}$ as a
function of distance $\tilde{d}_{\rm CL}$ for the bending rigidity ratios
$\kappa_{\rm v}/\kappa_{\rm p}=0.2,1.0$ and $5.0$ at fixed $v=0.8$, and
$\tilde{\sigma}=0.5$. Inset shows the same on log-log scale. The solid line
presents a power law with an exponent $-2$. The individual energy
contributions: the change in bending energy of the vesicle,
$\Delta\tilde{E}_{\rm b,v}$, the change in bending energy of the planar
membrane, $\Delta\tilde{E}_{\rm b,p}$, the change in surface energy of the
planar membrane, $\Delta\tilde{E}_{\rm s,p}$ and the change in adhesion
energy, $\Delta\tilde{E}_{\rm w}$, as function of distance $\tilde{d}_{\rm
CL}$ are presented for (b) $\kappa_{\rm v}/\kappa_{\rm p}=5.0$ (c)
$\kappa_{\rm v}/\kappa_{\rm p}=1.0$ and (d) $\kappa_{\rm v}/\kappa_{\rm
p}=0.2$.

Membrane-mediated interactions between two partial-wrapped vesicles. The
deformation of the initially planar, free membrane induces membrane-mediated
interactions between two partial-wrapped vesicles. When the vesicles approach
each other, both the initially planar membrane and the attached vesicles
deform. We predict membrane-mediated pair interactions between shallow- and
deep-wrapped prolate vesicles with $v=0.8$ at initially planar membranes with
a small finite membrane tension $\tilde{\sigma}=0.5$. The value of the
adhesion strength $\tilde{w}$ is chosen such that either a deep-wrapped or a
shallow-wrapped state is stable. The interaction potentials are obtained by
calculating the total energies of the system for various distances between the
vesicles. The change in the total energy $\Delta\tilde{E}$ is measured with
respect to the vesicles at infinite distance, i.e.,
$\Delta\tilde{E}=\tilde{E}(r)-\tilde{E}(\infty)$. We calculate the total-
energy difference $\Delta\tilde{E}$ as function of the reduced distance
$\tilde{d}_{\rm CL}=d_{\rm CL}/a_{0}$, where $d_{\rm CL}$ is the minimum
contactline-to-contactline distance between the two vesicles, and
$a_{0}=\sqrt{A_{v}/(4\pi)}$ is the radius of a spherical vesicle with the same
membrane area. To understand the origin of the membrane-mediated interaction,
we split the total potential energy $\Delta\tilde{E}$ into its individual
components,

$\Delta\tilde{E}(\tilde{d}_{\rm CL})=\Delta\tilde{E}_{\rm b,v}(\tilde{d}_{\rm
CL})+\Delta\tilde{E}_{\rm b,p}(\tilde{d}_{\rm CL})+\Delta\tilde{E}_{\rm
s,p}(\tilde{d}_{\rm CL})+\Delta\tilde{E}_{w}(\tilde{d}_{\rm CL}),$ (2)

where $\Delta\tilde{E}_{\rm b,v}$ is the change of bending energy of the
vesicles, $\Delta\tilde{E}_{\rm b,p}$ is the change of bending energy of the
planar membrane, $\Delta\tilde{E}_{\rm s,p}$ is the change of tension energy
of the planar membrane, and $\Delta\tilde{E}_{w}$ is the change of the
adhesion energy.

For two deep-wrapped vesicles, $\tilde{w}=15$, the energy $\Delta\tilde{E}$
increases monotonically with decreasing distance between the two vesicles, see
Fig. 5(a); the vesicles mutually repel each other. We find a purely repulsive
interaction with decreasing strength for decreasing bending rigidity of the
vesicle membrane. Within the considered distance range, the interaction
potential is well described by an effective power law with exponent $-2$, see
inset of Fig. 5(a). In Fig. 5(b)-(d), all energy components of Eq. (2) are
plotted as a function of $\tilde{d}_{\rm CL}$ for $\kappa_{\rm v}/\kappa_{\rm
p}=0.2$, $1$, and $5$. The bending-energy contributions, $\Delta\tilde{E}_{\rm
b,p}$ and $\Delta\tilde{E}_{\rm b,v}$, both increase with decreasing distance
between the vesicles. The deformation of the planar membrane increases as the
vesicles become stiffer, thus the contribution of $\Delta\tilde{E}_{\rm b,p}$
to the total energy increases with increasing $\kappa_{\rm v}/\kappa_{\rm p}$.
Within the range of our parameters, the relative contribution of
$\Delta\tilde{E}_{\rm b,v}$ increases as the vesicles become softer, because
the deformation of the vesicles becomes less expensive than that of the free
membrane. The contribution of the adhesion energy $\Delta\tilde{E}_{w}$ is
significant for all cases. We find that $\Delta\tilde{E}_{w}$ becomes more
negative at short distances, which is expected as the adhered area increases
at short distances; the change of wrapping fraction becomes less important
with decreasing $\kappa_{\rm v}/\kappa_{\rm p}$, see Fig. S12 in the
Supporting Information. Thus, the contribution of $\Delta\tilde{E}_{w}$
becomes weaker with decreasing $\kappa_{\rm v}/\kappa_{\rm p}$. For all cases,
the repulsive potential mainly originates from $\Delta\tilde{E}_{\rm b,p}$ and
$\Delta\tilde{E}_{\rm b,v}$. We observe a weak increase of
$\Delta\tilde{E}_{\rm s,p}$ as the distance between the vesicles decreases,
which is connected to the increase of adhered membrane area as the vesicles
approach each other, see Fig. S12 in the Supporting Information. The
contribution of $\Delta\tilde{E}_{\rm s,p}$ to the total energy is negligible
because a very small membrane tension is applied.

Figure 6: Shallow-wrapped vesicles at $\tilde{w}=10$. (a) Cross-section side
view of two vesicles with side-by-side (SS) and tip-to-tip (TT) orientations
at different distances for the choice of parameters $v=0.8$,
$\tilde{\sigma=0.5}=0.5$, and $\kappa_{\rm v}/\kappa_{\rm p}=5.0$. (b)
Interaction potential $\Delta\tilde{E}$, as a function of the distance
$\tilde{d}_{\rm CL}$ between the two vesicles in shallow-wrapped states with
SS- and TT-orientations. For SS orientation the interaction potential is
plotted for the bending rigidity ratios $\kappa_{\rm v}/\kappa_{\rm
p}=0.2,1.0$ and $5.0$ at $v=0.8$ and $\tilde{\sigma}=0.5$. The inset shows the
same on a log-log scale. The solid line presents a power law with an exponent
$-2.5$. For TT orientation the interaction potential is shown for $\kappa_{\rm
v}/\kappa_{\rm p}=5.0$. The individual energy contributions: the change in
bending energy of the vesicle, $\Delta\tilde{E}_{\rm b,v}$, the change in
bending energy of the planar membrane, $\Delta\tilde{E}_{\rm b,p}$, the change
in surface energy of the planar membrane, $\Delta\tilde{E}_{\rm s,p}$ and the
change in adhesion energy, $\Delta\tilde{E}_{\rm w}$, as function of distance
$\tilde{d}_{\rm CL}$ are presented at $\kappa_{\rm v}/\kappa_{\rm p}=5.0$ and
$\tilde{\sigma}=0.5$ for (c) SS-orientation and (d) TT-orientation.

We also investigate the membrane-mediated interaction between two shallow-
wrapped vesicles with reduced volume $v=0.8$ and tension $\tilde{\sigma}=0.5$
for the planar membrane, see Fig. 6(a). The adhesion strength is fixed at
$\tilde{w}=10$, such that the shallow-wrapped state is stable for all
considered values of $\kappa_{\rm v}/\kappa_{\rm p}$. For side-by-side (SS)
orientation, the interaction is purely repulsive for $\kappa_{\rm
v}/\kappa_{\rm p}=0.2,1$, and $5$, see Fig. 6(b). The interaction strength
decreases as the vesicles become softer, similar as for deep-wrapped vesicles.
A qualitative change in the interaction potential is associated with the shape
change of the vesicles from prolate for $\kappa_{\rm v}/\kappa_{\rm p}=5$ to
oblate for $\kappa_{\rm v}/\kappa_{\rm p}\lesssim 1$. The shape change also
leads to higher wrapping fractions of the vesicles from $f_{w}\approx 0.14$
for $\kappa_{\rm v}/\kappa_{\rm b}=5$ to $f_{w}\approx 0.27$ for $\kappa_{\rm
v}/\kappa_{\rm b}=1$. Similarly, the deformation energy of the planar membrane
decreases more as the prolate vesicles approach each other in side-by-side
orientation than for the oblate shapes. However, the deformation energy of the
free planar membrane, without the adhered part of the planar membrane,
increases with increasing $\kappa_{\rm v}/\kappa_{\rm p}$, see Fig. S11 in the
Supporting Information.

For SS orientation and $\kappa_{\rm v}/\kappa_{\rm p}=5$, the high curvature
of the free membrane between the particles increases the bending-energy cost
for the free membrane, see Fig. S11 in the Supporting Information. This leads
to partial unwrapping of the vesicles from the planar membrane, see Fig. 6(c)
and Fig. S12 in the Supplementary Information. We find both bending-energy
contributions, $\Delta\tilde{E}_{\rm b,p}$ and $\Delta\tilde{E}_{\rm b,v}$, to
become more negative with decreasing distance. The change of adhesion energy
$\Delta\tilde{E}_{w}$ is increasing with decreasing vesicle-vesicle distance,
which is the dominating contribution causing the repulsion between the
vesicles. The contribution of $\Delta{\tilde{E}}_{\rm s,p}$ to the total
energy decreases with decreasing $\tilde{d}_{\rm CL}$, which is expected
because the total membrane area decreases as two vesicles approach each other.
However, the contribution of $\Delta{\tilde{E}}_{\rm s,p}$ to the total energy
is negligible. The total energy as function of the distance can be described
by an effective power law $\tilde{d}_{\rm CL}^{-2.5}$, see the inset of Fig.
5(b). Thus, a smaller area of the vesicles is adhered to the planar membrane
as they approach each other, opposite to deep-wrapped vesicles where the
wrapping fraction increases. As a result, a qualitative change is observed in
the behavior of the individual components in comparison with deep-wrapped
vesicles. Our shallow-wrapped vesicles in SS orientation experience stiffer
membrane-mediated interaction potentials than our deep-wrapped vesicles.

For tip-to-tip (TT) orientation and $\kappa_{\rm v}/\kappa_{\rm p}=5$, a
cooperative deformation of the membrane by the two vesicles leads to the
formation of a joint "trough" for both particles without a strong increase of
the deformation-energy cost of the free membrane, see Fig. 6(a) and Fig.
7(e,f). The interaction is weakly attractive. To understand this behavior, we
again look into the individual energy components, see Fig. 6(d). In contrast
to the SS orientation, here we find that the wrapping fraction of the vesicles
increases as the vesicles approach each other. As a result, tension- and
bending-energy contributions for both the planar membrane and the vesicles
increase as the distance between the vesicles decreases. The effective
attraction originates from the gain in adhesion energy. For shallow-wrapped
states and $\kappa_{\rm v}/\kappa_{\rm p}\lesssim 1$ the vesicle shape changes
from prolate to oblate; therefore, there is no distinction between TT and SS
orientations.

Now, we discuss shapes and local bending energies of the free membrane for
both, deep-wrapped and shallow-wrapped vesicles, see Fig. 7(a)-(f). At the
contactline, where the free membrane detaches from the vesicle, the membrane
height deviates most from the height of the wire frame. For two deep-wrapped
vesicles with contactline-contactline distance $\tilde{d}_{\rm CL}=0.97$, the
free-membrane height also strongly deviates from the wire frame height in the
middle between the vesicles, see Fig. 7(a). Here, the deformation is strong
because the membrane aims to decrease the high local bending-energy cost, see
Fig. 7(b); yet, this energy cost causes the repulsive interaction between the
vesicles, compare Fig. 5(b). The repulsion leads to tilting of the vesicles,
see Fig. S7 in the Supplementary Information. Interestingly, the membrane-
mediated interaction induces an increase of the center-of-mass height of the
vesicles with respect to the wire frame, see Fig. S8 in the Supplementary
Information.

For two shallow-wrapped vesicles with contactline-contactline distance
$\tilde{d}_{\rm CL}=1.36$, the height of the free membrane deviates most
strongly from the wire frame height at the tips of the vesicles, see Fig.
7(c). The bending energy cost between the vesicles is very high for SS
orientation, see Fig. 7(d), which leads to a partial unwrapping of the vesicle
sides that face each other and therefore a repulsive interaction because of
the decreasing adhesion energy when the vesicles approach each other, see Fig.
6(c) and Fig. S12 in the Supplementary Information. For TT orientation, the
bending energy of the membrane between the vesicles is very small although the
membrane is also strongly deformed, see Fig. 7(e). The membrane height
deviates strongest from the wire-frame height at the tips that face each
other–and wrapping both vesicles together increases the wrapping at the sides
as well. However, the overall increase of the bending energy in TT orientation
is much lower than in SS orientation; the joint deformation of the membrane by
both vesicles strongly increases the wrapping at their sides and thereby the
adhesion energy $E_{w}$, see Fig. 7(f) and Fig. 6(d). Also for shallow-wrapped
vesicles, the heights of the centers of mass of the vesicles decrease with
decreasing distance, see Fig. S8(b) in the Supporting Information.

Figure 7: Free-membrane deformation for two vesicles at a planar membrane.
(a,c,e) Local membrane height difference $\Delta\tilde{h}=[h-h_{0}]/a_{0}$
with the local membrane height $h$ and the height $h_{0}$ of the wire frame
spanning the membrane patch, and (b,d,f) local bending energy of the free
membrane. The values are shown in the $x-y$ plane for vesicles with $v=0.8$,
reduced membrane tension $\tilde{\sigma}=0.5$, and $\kappa_{\rm v}/\kappa_{\rm
p}=5.0$. (a, b) Deep-wrapped vesicles with their centers of mass at
$(x_{1},y_{1})=(-x_{2},y_{2})=(0.49,0)$, (c,d) shallow-wrapped in side-by-side
orientation with their centers of mass at
$(x_{1},y_{1})=(-x_{2},y_{2})=(0.68,0)$, and (e, f) shallow-wrapped in tip-to-
tip orientation with their centers of mass at
$(x_{1},y_{1})=(-x_{2},y_{2})=(0.75,0)$.

## Conclusions

In this work, we have predicted the wrapping behavior and calculated membrane-
mediated interactions between prolate vesicles at planar membranes. For
single-vesicle systems, we systematically varied the reduced volume $v$ of the
vesicles, the bending rigidity ratio $\kappa_{\rm v}/\kappa_{\rm p}$, and the
membrane tension $\tilde{\sigma}$ of the planar membrane. Wrapping always
starts from the minimum curvature regions of the vesicles. Thus, initial
wrapping for low adhesion strengths occurs in submarine orientation where the
major axis of the vesicle is parallel to the planar membrane. With increasing
adhesion strength, for stiff vesicles the orientation changes from submarine
to rocket, where the major axis of the vesicle is perpendicular to the planar
membrane. For soft vesicles, a shape and orientation change of partial-wrapped
vesicles from prolate submarine to oblate to prolate rocket can be observed.
The binding transition, the shallow-to-deep-wrapped transition, and the
envelopment transition are located and characterized by systematically varying
the adhesion strength. We have shown that the binding and the shallow-to-deep-
wrapped transition are always discontinuous, and the envelopment transition is
always continuous. For fixed $v$ and $\tilde{\sigma}$, we have shown that the
binding transition shifts to lower adhesion strengths and the envelopment
transition shifts to higher adhesion strengths with decreasing bending-
rigidity ratio $\kappa_{\rm v}/\kappa_{\rm p}$. The softer the vesicle, the
easier it attaches to the membrane, but the more difficult it gets completely
wrapped–which is a generic behavior of soft, deformable particles. For fixed
$v$ and $\kappa_{\rm v}/\kappa_{\rm p}$, we have shown that the binding
transition is independent of the membrane tension. An increase of membrane
tension stabilizes shallow-wrapped states over deep-wrapped and complete-
wrapped states.

The membrane-mediated interaction between two shallow-wrapped or two deep-
wrapped prolate vesicles ($v=0.8$) at planar membranes is studied by
systematically varying the distance between two partial-wrapped vesicles.
Particle systems are more complex than spherical-cap inclusions42, 30, 43, 44,
45, curved membrane domains 46, and curved proteins47 for which the wrapping
fraction does not depend on the distance. For deep-wrapped states of our
prolate vesicles, we found the interaction potential to be repulsive, which
originates from the deformation of both, free membrane and vesicles. The
strength of the repulsive interaction decreases with the decreasing
$\kappa_{\rm v}/\kappa_{\rm p}$, i.e., with increasing softness of the
vesicles. For shallow-wrapped states, a qualitative change in the interaction
potential from attractive in tip-to-tip orientation to repulsive in side-by-
side orientation is observed. For side-by-side orientation, the interaction
potential is purely repulsive and becomes weaker as the vesicles become
softer, which is associated with the shape change of the vesicles from prolate
to oblate. Although, the interaction for both, deep-wrapped vesicles and
shallow-wrapped vesicles in SS orientation, is repulsive, the reason is
different in both cases. For side-by-side orientation, the repulsive
interaction originates from the loss of adhesion energy as the two vesicles
approach each other. For tip-to-tip orientation, the interaction energy is
attractive, which originates from the gain in adhesion energy as the vesicles
approach each other.

Our theoretical predictions for wrapping and for pair-interactions of soft
particles at membranes can be used to optimise the shapes and the elastic
properties of deformable particles for efficient use in nanomedicine, such as
for targeted drug delivery. It will be also interesting to investigate
membrane-mediated interactions between many particles, which will help us to
understand more complex problems like aggregation of virions on lipid-bilayer
membranes. The understanding of the wrapping behavior of the vesicle-membrane
systems is leading off to investigate the interaction of polymeric particles,
e.g., star polymers, polymer-grafted nanoparticles, and microgels, with lipid
bilayer membranes. However, contrary to vesicles polymeric particles have bulk
and shear elasticities, and their surface area increases upon binding to a
membrane 48. Yet the knowledge that we gained from non-spherical vesicle-
membrane systems provides a starting point to estimate the membrane-mediated
interaction for anisotropic soft particles in general.

##  2 Model and methods

The continuum model in Eq. (1) is limited to systems with particle sizes that
are at least few times larger than the thickness of a lipid-bilayer membranes;
bending and tension energy both contribute in the wrapping process. The total
energy of the system can be expressed as a dimensionless quantity by dividing
by the wrapping energy of spherical particles, $\pi\kappa_{\rm p}$, and
expressing all length scales in terms of the membrane area $A_{\rm v}$ of the
vesicle. This gives

$\tilde{E}=\frac{E}{\pi\kappa_{\rm p}}=\frac{2}{\pi A_{\rm
v}}\left[\int_{A_{\rm p}}dS[A_{\rm
v}H^{2}+\pi\tilde{\sigma}]+\frac{\kappa_{\rm v}}{\kappa_{\rm p}}\int_{A_{\rm
v}}dSA_{\rm v}H^{2}\right]-\tilde{w}\int_{A_{\rm ad}}\frac{dS}{A_{\rm v}}\,,$
(3)

where $\tilde{\sigma}=\sigma A_{\rm v}/(2\pi\kappa_{\rm p})$ and
$\tilde{w}=wA_{\rm v}/(\pi\kappa_{\rm p})$ are the reduced surface tension and
reduced adhesion strength, respectively.

The integrals are discretized using triangulated membranes 49, 50, 51. For
energy minimization, we employ the freely available program package “Surface
Evolver”, where a discretized surface is formed of vertices, edges, and facets
52. To calculate minimal-energy shapes and associated deformation energies, we
refine the triangulation and minimize the energy at each refinement level in
an alternating sequence. The refinement of the triangles and energy
minimization continue until the desired accuracy is achieved. The enclosed
volumes ($V_{\rm v}$) and vesicle areas ($A_{\rm v}$) are fixed with the help
of Lagrange multipliers. The vesicle shapes are controlled via the reduced
volume $v$. The upper bound $v=1$ corresponds to a spherical vesicle, while
$v<1$ represents nonspherical vesicles, see Fig. 1S in the Supporting
Information.

The numerical data for the deformation energy are fitted using a piecewise
function, independently for submarine and rocket orientation. In each region,
we fit the energy using a fourth-order polynomial

$f(f_{w})=\sum_{i=0}^{4}a_{0}[f_{w}]^{i}$ (4)

with the coefficients $a_{0},...,a_{4}$. For the calculation of the phase
diagrams we use the sum of deformation and adhesion energy, where the latter
is proportional to the wrapping fraction. To locate different transitions,
i.e. binding transitions, shallow-wrapped to deep-wrapped transitions, and
envelopment transitions, we take the first derivative of the deformation
energy with respect to the wrapping fraction for different adhesion strengths.
The zeros of the first derivative indicate the local minima and maxima of the
wrapping energies.

To calculate membrane-mediated pair interaction potentials, two vesicles are
placed on the planar membrane at fixed center-of-mass distances
$\tilde{d}_{\rm CC}=d_{\rm CC}/a_{0}$, where $a_{0}=\sqrt{(A_{\rm v}/4\pi)}$.
However, for the presentation of the results, we use the contactline-to-
contactline distance $\tilde{d}_{\rm CL}=d_{\rm CL}/a_{0}$; we have also
calculated the vesicle surface-to-surface distances $\tilde{d}_{\rm SS}=d_{\rm
SS}/a_{0}$, see Table S2 in the Supporting Information. For each distance, the
total energy is minimized and the deformation energy $\Delta\tilde{E}$ as
function of the wrapping fraction $f_{w}$ is obtained. We exploit the mirror
symmetry and calculate only a quarter or half of the actual system. For the
deep-wrapped states, the contactline is stabilized by forcing it to a plane
with an arbitrary tilt angle. As for the single-vesicle systems, the
deformation energy is fit using fourth-order polynomials. The adhesion
strength $\tilde{w}$ is chosen such that the minimum of $\Delta\tilde{E}$ is
either a stable deep-wrapped or a stable shallow-wrapped state for all
vesicle-vesicle distances.

We gratefully acknowledge financial support from the Deutsche
Forschungsgemeinschaft (DFG) within the SFB 985 “Functional Microgels and
Microgel Systems” and helpful discussions with Jerome Crassous (Aachen) and
Ken Brakke (Selinsgrove).

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$\displaystyle\qquad+\sqrt{s}\|\frac{1}{n}(\Sigma^{1/2})_{JJ}(Y_{J}^{\top}Y_{J}-nI)(\Sigma^{1/2})_{JJ}U_{\perp}\|_{\max}\|U_{\perp}^{\top}\tilde{U}_{J}\|$
$\displaystyle\lesssim\lambda_{1}\sqrt{\frac{k\log(p)}{n}}+\lambda_{1}\sqrt{\frac{s\log(p)}{n}}\|U_{\perp}^{\top}\tilde{U}_{J}\|.$

So what remains is to bound the term $\|U_{\perp}^{\top}\tilde{U}_{J}\|$,
However, we note that this is simply (by a factor of $\sqrt{2}$) the
$\sin\Theta$ distance between the subspace $U_{J}U_{J}^{\top}$ and
$\tilde{U}_{J}\tilde{U}_{J}^{\top}$ (see Lemma 10 in Appendix C). Therefore,
by Proposition 1, we have that this can be bounded by

$\displaystyle\|U_{\perp}^{\top}\tilde{U}_{J}\|\lesssim\frac{\lambda_{1}}{\lambda_{k}-\lambda_{k+1}}\sqrt{\frac{s\log(p)}{n}}.$

Putting it all together, this yields that with probability at least
$1-O(p^{-4})$ that

$\displaystyle\|K_{1}\|_{2\to\infty}$
$\displaystyle=\|\frac{1}{n}(\Sigma^{1/2})_{JJ}(Y_{J}^{\top}Y_{J}-nI)(\Sigma^{1/2})_{JJ}\tilde{U}_{J}\|_{2\to\infty}$
$\displaystyle\lesssim\frac{\lambda_{1}^{2}}{\lambda_{k}-\lambda_{k+1}}\frac{s\log(p)}{n}+\lambda_{1}\sqrt{\frac{k\log(p)}{n}},$

which is the desired bound. ∎

Again, we repeat the statement of Lemma 8.

See 8

Recall that

$\displaystyle K_{2}:$
$\displaystyle=\Sigma_{JJ^{c}}^{1/2}Y_{J^{c}}^{\top}Y_{J}(\Sigma^{1/2})_{JJ}\tilde{U}_{J}$

###### Proof of Lemma 8.

We have that

$\displaystyle\|\frac{1}{n}\Sigma_{JJ^{c}}^{1/2}Y_{J^{c}}^{\top}Y_{J}(\Sigma^{1/2})_{JJ}\tilde{U}_{J}\|_{2\to\infty}$
$\displaystyle\leq\|\frac{1}{n}\Sigma_{JJ^{c}}^{1/2}Y_{J^{c}}^{\top}Y_{J}(\Sigma^{1/2})_{JJ}U_{J}\|_{2\to\infty}$
$\displaystyle\qquad+\|\frac{1}{n}\Sigma_{JJ^{c}}^{1/2}Y_{J^{c}}^{\top}Y_{J}(\Sigma^{1/2})_{JJ}U_{\perp}\|_{2\to\infty}\|U_{\perp}^{\top}\tilde{U}_{J}\|$
$\displaystyle\leq\sqrt{k}\|\frac{1}{n}\Sigma_{JJ^{c}}^{1/2}Y_{J^{c}}^{\top}Y_{J}(\Sigma^{1/2})_{JJ}U_{J}\|_{\max}$
$\displaystyle\qquad+\sqrt{s}\|\frac{1}{n}\Sigma_{JJ^{c}}^{1/2}Y_{J^{c}}^{\top}Y_{J}(\Sigma^{1/2})_{JJ}U_{\perp}\|_{\max}\|U_{\perp}^{\top}\tilde{U}_{J}\|.$
(27)

We bound each norm inside the max separately. Define the random variable
$\eta_{ij}$ as the $i,j$ entry of the matrix
$\Sigma_{JJ^{c}}^{1/2}Y_{J^{c}}^{\top}Y_{J}(\Sigma^{1/2})_{JJ}U_{J}$. Then

$\displaystyle\eta_{ij}$
$\displaystyle=\frac{1}{n}\sum_{q=1}^{n}\sum_{k=1}^{s}\sum_{l=1}^{p-s}[\Sigma_{JJ^{c}}^{1/2}]_{il}\xi^{(q)}_{s+l,k}\bigg{(}(\Sigma^{1/2})_{JJ}U_{J}\bigg{)}_{kj},$

where $\xi^{(q)}_{s+l,k}:=Y_{q,s+l}Y_{qk}$. Following a strategy similar to
the proof of Lemma 6, we have to bound both the maximum and sum of squared
$\psi_{1}$ norms of the random variable

$\displaystyle\alpha_{qlj}:=\frac{1}{n}[\Sigma_{JJ^{c}}^{1/2}]_{il}\xi^{(q)}_{s+l,k}\bigg{(}(\Sigma^{1/2})_{JJ}U_{J}\bigg{)}_{kj}.$

The squared entries satisfy

$\displaystyle\|\frac{1}{n}[\Sigma_{JJ^{c}}^{1/2}]_{il}\xi^{(q)}_{s+l,k}\bigg{(}(\Sigma^{1/2})_{JJ}U_{J}\bigg{)}_{kj}\|_{\psi_{1}}^{2}$
$\displaystyle\leq\frac{1}{n^{2}}([\Sigma_{JJ^{c}}^{1/2}]_{il})^{2}\bigg{(}(\Sigma^{1/2})_{JJ}U_{J}\bigg{)}_{kj}^{2}.$

Summing up over $q,l,j$,

$\displaystyle\sum_{q=1}^{n}\sum_{k=1}^{s}\sum_{l=1}^{p-s}\|\alpha_{qlj}\|_{\psi_{1}}^{2}$
$\displaystyle\leq\frac{1}{n}\sum_{k=1}^{s}\sum_{l=1}^{p-s}([\Sigma_{JJ^{c}}^{1/2}]_{il})^{2}\bigg{(}(\Sigma^{1/2})_{JJ}U_{J}\bigg{)}_{kj}^{2}$
$\displaystyle\leq\frac{1}{n}\sum_{k=1}^{s}\bigg{(}(\Sigma^{1/2})_{JJ}U_{J}\bigg{)}_{kj}^{2}\|\Sigma_{JJ^{c}}^{1/2}\|_{2\to\infty}^{2}$
$\displaystyle\leq\frac{\lambda_{1}\|\Sigma_{JJ^{c}}^{1/2}\|_{2\to\infty}^{2}}{n}.$

Also,

$\displaystyle\max_{q,l,j}\|\alpha_{qlj}\|_{\psi_{1}}$
$\displaystyle\leq\frac{1}{n}\sqrt{\lambda_{1}}\|\Sigma_{JJ^{c}}^{1/2}\|_{2\to\infty}.$

By the the Generalized Bernstein inequality (Theorem 3 in Appendix C),

$\displaystyle\mathbb{P}\bigg{(}|\eta_{ij}|>t\bigg{)}\leq
2\exp\bigg{(}-cn\min\bigg{[}\frac{t^{2}}{\lambda_{1}\|\Sigma_{JJ^{c}}^{1/2}\|_{2\to\infty}^{2}},\frac{t}{\sqrt{\lambda_{1}}\|\Sigma_{JJ^{c}}^{1/2}\|_{2\to\infty}}\bigg{]}\bigg{)}.$

Again taking
$t=C\|\Sigma_{JJ^{c}}^{1/2}\|_{2\to\infty}\sqrt{\lambda_{1}}\sqrt{\frac{\log(s)+\log(k)+4\log(p)}{n}}$
shows that this holds with probability $1-O(s^{-1}k^{-1}p^{-4})$. Taking a
union over all $s$ rows and $k$ columns of the matrix yields that

$\displaystyle\|\frac{1}{n}\Sigma_{JJ^{c}}^{1/2}Y_{J^{c}}^{\top}Y_{J}(\Sigma^{1/2})_{JJ}U_{J}\|_{\max}$
$\displaystyle\lesssim\|\Sigma_{JJ^{c}}^{1/2}\|_{2\to\infty}\sqrt{\lambda_{1}}\sqrt{\frac{\log(p)}{n}}$
$\displaystyle\lesssim\lambda_{1}\sqrt{\frac{\log(p)}{n}}.$

Applying precisely the same argument to the other term yields with probability
$1-O(p^{-4})$ that

$\displaystyle\|\frac{1}{n}\Sigma_{JJ^{c}}^{1/2}Y_{J^{c}}^{\top}Y_{J}(\Sigma^{1/2})_{JJ}U_{\perp}\|_{\max}$
$\displaystyle\lesssim\lambda_{1}\sqrt{\frac{\log(p)}{n}}.$

Therefore, combining these bounds with the initial bound in (27) and
Proposition 1 and the equivalent expressions for the $\sin\Theta$ distances
(Lemma 10 in Appendix C), we have that with probability at least
$1-O(p^{-4})$,

$\displaystyle\|K_{2}\|_{2\to\infty}=\|\frac{1}{n}\Sigma_{JJ^{c}}^{1/2}Y_{J^{c}}^{\top}Y_{J}(\Sigma^{1/2})_{JJ}\tilde{U}_{J}\|_{2\to\infty}$
$\displaystyle\leq\sqrt{k}\|\frac{1}{n}\Sigma_{JJ^{c}}^{1/2}Y_{J^{c}}^{\top}Y_{J}(\Sigma^{1/2})_{JJ}U_{J}\|_{\max}$
$\displaystyle\qquad+\sqrt{s}\|\frac{1}{n}\Sigma_{JJ^{c}}^{1/2}Y_{J^{c}}^{\top}Y_{J}(\Sigma^{1/2})_{JJ}U_{\perp}\|_{\max}\|U_{\perp}^{\top}\tilde{U}_{J}\|$
$\displaystyle\lesssim\lambda_{1}\sqrt{\frac{k\log(p)}{n}}+\frac{\lambda_{1}^{2}}{\lambda_{k}-\lambda_{k+1}}\frac{s\log(p)}{n}$

as desired. ∎

###  B.6 Proof of Lemma 9

It will be useful to collect some properties of the matrix
$\Sigma_{JJ^{c}}^{1/2}$, which we state as a proposition.

######  Proposition 2 (Properties of the Matrix $\Sigma_{JJ^{c}}^{1/2}$).

The matrix $\Sigma_{JJ^{c}}^{1/2}$ satisfies

$\displaystyle\|\Sigma_{JJ^{c}}^{1/2}\|$ $\displaystyle\leq
2\sqrt{\lambda_{1}}$

Furthermore, the left singular subspace of $\Sigma^{1/2}_{JJ^{c}}$ must
contain columns of $U_{\perp}$.

###### Proof of Proposition 2.

First, we note that

$\displaystyle\|\Sigma_{JJ^{c}}^{1/2}\|$
$\displaystyle=\bigg{\|}\begin{pmatrix}0&(\Sigma^{1/2})_{JJ^{c}}\\\
((\Sigma^{1/2})_{JJ^{c}})^{\top}&0\end{pmatrix}\bigg{\|}$
$\displaystyle\leq\|\Sigma\|^{1/2}+\bigg{\|}\begin{pmatrix}(\Sigma^{1/2})_{JJ}&0\\\
0&\Sigma_{J^{c}J^{c}}^{1/2}\end{pmatrix}\bigg{\|}$ $\displaystyle\leq
2\sqrt{\lambda_{1}},$

since eigenvalues bound eigenvalues of any principal submatrix. For the second
claim, note that

$\displaystyle\Sigma^{1/2}\begin{pmatrix}U_{J}\\\ 0\end{pmatrix}$
$\displaystyle=\begin{pmatrix}(\Sigma^{1/2})_{JJ}&(\Sigma^{1/2})_{JJ^{c}}\\\
((\Sigma^{1/2})_{JJ^{c}})^{\top}&\Sigma_{J^{c}J^{c}}^{1/2}\end{pmatrix}\begin{pmatrix}U_{J}\\\
0\end{pmatrix}$ $\displaystyle=\begin{pmatrix}U_{J}\\\
0\end{pmatrix}\Lambda^{1/2}.$

This shows that the matrix $(\Sigma^{1/2}_{JJ^{c}})^{\top}$ satisfies
$(\Sigma^{1/2}_{JJ^{c}})^{\top}U_{J}=0$, so that its null space must contain
the space spanned by $U_{J}$. However, this also shows that since
$(\Sigma^{1/2}_{JJ^{c}})^{\top}\in\mathbb{R}^{(p-s)\times s}$, then its rank
is at most $s-k$. Hence, define
$(\Sigma^{1/2}_{JJ^{c}})^{\top}=V_{1}DV_{2}^{\top}$ as the reduced singular
value decomposition of $(\Sigma^{1/2}_{JJ^{c}})^{\top}$. Since its rank is at
most $s-k$, we have that $V_{1}\in\mathbb{O}(p-s,s-k)$,
$V_{2}\in\mathbb{O}(s,s-k)$, and $D$ is an $s-k\times s-k$ diagonal matrix of
singular values.

Since $(\Sigma^{1/2}_{JJ^{c}})^{\top}U_{J}=V_{1}DV_{2}^{\top}U_{J}=0$, the
term $V_{2}\in\mathbb{O}(s,s-k)$ must span a space perpendicular to $U_{J}$.
The only matrix up to choice of basis in $\mathbb{O}(s,s-k)$ satisfying
$V_{2}^{\top}U_{J}=0$ is the matrix $U_{\perp}$, which establishes the second
claim. ∎

Therefore, all this shows that

  * •

The left singular subspace of $\Sigma^{1/2}_{JJ^{c}}$ contains columns of
$U_{\perp}$;

  * •

Its singular values are all uniformly bounded by $2\sqrt{\lambda_{1}}$.

We are now prepared to prove Lemma 9. See 9

###### Proof of Lemma 9.

Let $\Sigma_{JJ^{c}}^{1/2}$ have singular value decomposition
$U_{\perp}DV^{\top}$, where $U_{\perp}\in\mathbb{O}(s,s-k)$, $D_{ii}\geq 0$,
$1\leq i\leq s-k$, $V\in\mathbb{O}(p-s,s-k)$. We will show the result for
$D_{ii}>0$, though the same proof goes through if $D_{ii}=0$ for some $i$.

Then the term $K_{3}$ satisfies

$\displaystyle\|K_{3}\|_{2\to\infty}$
$\displaystyle=\|(\Sigma^{1/2})_{JJ}\frac{Y_{J}^{\top}Y_{J^{c}}}{n}(\Sigma_{JJ^{c}}^{1/2})^{\top}\tilde{U}_{J}\|_{2\to\infty}$
$\displaystyle\leq\|(\Sigma^{1/2})_{JJ}\frac{Y_{J}^{\top}Y_{J^{c}}}{n}VDU_{\perp}^{\top}\tilde{U}_{J}\|_{2\to\infty}$
$\displaystyle\leq\|(\Sigma^{1/2})_{JJ}\frac{Y_{J}^{\top}Y_{J^{c}}}{n}VDU_{\perp}^{\top}U_{\perp}\|_{2\to\infty}\|U_{\perp}^{\top}\tilde{U}_{J}\|$
$\displaystyle\leq\|(\Sigma^{1/2})_{JJ}\frac{Y_{J}^{\top}Y_{J^{c}}}{n}V\|_{2\to\infty}\sqrt{\lambda_{1}}\|U_{\perp}^{\top}\tilde{U}_{J}\|.$
(28)

The term $\|U_{\perp}^{\top}\tilde{U}_{J}\|$ can be bounded via Proposition 1
and Lemma 10 in Appendix C. So what remains is to bound the $2\to\infty$ norm
in (28). Note that the matrix $V$ is of column dimension at most $(s-k)$.
Hence, each of the $s$ rows of the matrix $(\Sigma^{1/2})_{JJ}Y_{J}Y_{J^{c}}V$
is of dimension at most $s-k$.

Following a strategy similar to that in Lemmas 7 and 8, we have that

$\displaystyle\|(\Sigma^{1/2})_{JJ}\frac{Y_{J}^{\top}Y_{J^{c}}}{n}V\|_{2\to\infty}$
$\displaystyle\leq\sqrt{s-k}\max_{i,j}\bigg{|}(\Sigma^{1/2})_{JJ}\frac{Y_{J}^{\top}Y_{J^{c}}}{n}V\bigg{|}_{i,j}$
$\displaystyle\leq\sqrt{s}\max_{i,j}\bigg{|}(\Sigma^{1/2})_{JJ}\frac{Y_{J}^{\top}Y_{J^{c}}}{n}V\bigg{|}_{i,j}.$

By analogous arguments as in Lemma 8, the $i,j$ entry is a sum of independent
mean-zero subexponential random variables, each with $\psi_{1}$ norm bounded
$\frac{1}{n}\sqrt{\lambda_{1}}$. Therefore, by Bernstein’s inequality, any
$i,j$ entry is bounded by

$\displaystyle C\sqrt{\lambda_{1}}\sqrt{\frac{\log(p)}{n}}$

with probability at most $1-O(p^{-3})$. Combining with Proposition 1, we have
the bound

$\displaystyle\|K_{3}\|_{2\to\infty}$
$\displaystyle\lesssim\lambda_{1}\sqrt{\frac{s\log(p)}{n}}\|U_{\perp}^{\top}\tilde{U}_{J}\|$
$\displaystyle\lesssim\lambda_{1}\sqrt{\frac{s\log(p)}{n}}\bigg{(}\frac{\lambda_{1}}{\lambda_{k}-\lambda_{k+1}}\sqrt{\frac{s\log(p)}{n}}\bigg{)}$
$\displaystyle\lesssim\frac{\lambda_{1}^{2}}{\lambda_{k}-\lambda_{k+1}}\frac{s\log(p)}{n}$

as desired.

For the term $K_{4}$, we see that

$\displaystyle\|K_{4}\|_{2\to\infty}$
$\displaystyle=\|\Sigma^{1/2}_{JJ^{c}}\bigg{(}\frac{Y_{J^{c}}^{\top}Y_{J^{c}}}{n}-I\bigg{)}VDU_{\perp}^{\top}\tilde{U}_{J}\|_{2\to\infty}$
$\displaystyle\leq\|\Sigma^{1/2}_{JJ^{c}}\bigg{(}\frac{Y_{J^{c}}^{\top}Y_{J^{c}}}{n}-I\bigg{)}V\|_{2\to\infty}\sqrt{\lambda_{1}}\|U_{\perp}^{\top}\tilde{U}_{J}\|$
$\displaystyle\leq\sqrt{s\lambda_{1}}\|\Sigma^{1/2}_{JJ^{c}}\bigg{(}\frac{Y_{J^{c}}^{\top}Y_{J^{c}}}{n}-I\bigg{)}V\|_{\max}\|U_{\perp}^{\top}\tilde{U}_{J}\|.$
(29)

We will bound the term inside the max norm for fixed $i$ and $j$. Observe that

$\displaystyle\bigg{|}\bigg{(}\Sigma^{1/2}_{JJ^{c}}\bigg{(}\frac{Y_{J^{c}}^{\top}Y_{J^{c}}}{n}-I\bigg{)}V\bigg{)}_{ij}\bigg{|}$
$\displaystyle=\max_{i,j}\bigg{|}\frac{1}{n}\sum_{q}\sum_{k,l}\bigg{(}\Sigma^{1/2}_{JJ^{c}}\bigg{)}_{ik}(Y_{qk}Y_{ql}-\mathbb{E}Y_{qk}Y_{ql})V_{lj}\bigg{|}$
$\displaystyle\leq\max_{q}\bigg{|}\sum_{k,l}\bigg{(}\Sigma^{1/2}_{JJ^{c}}\bigg{)}_{ik}(Y_{qk}Y_{ql}-\mathbb{E}Y_{qk}Y_{ql})V_{lj}\bigg{|}.$

We will first bound the term inside the absolute value for fixed $q$ by
Hanson-Wright (Theorem 4 in Appendix C). Let $M$ be the matrix defined via

$\displaystyle M_{kl}:=\bigg{(}\Sigma_{JJ^{c}}^{1/2}\bigg{)}_{ik}V_{lj}.$

Then

$\displaystyle\|M\|_{F}^{2}$
$\displaystyle=\sum_{k,l}\bigg{(}\Sigma_{JJ^{c}}^{1/2}\bigg{)}_{ik}^{2}V_{lj}^{2}=\sum_{k}\bigg{(}\Sigma_{JJ^{c}}^{1/2}\bigg{)}_{ik}^{2}\leq\|\Sigma_{JJ^{c}}^{1/2}\|_{2\to\infty}^{2}\leq
4\lambda_{1}.$

Therefore, by applying the Hanson-Wright inequality, for any fixed $q$ it
holds that

$\displaystyle\mathbb{P}\bigg{(}\bigg{|}\sum_{k,l}\bigg{(}\Sigma^{1/2}_{JJ^{c}}\bigg{)}_{ik}(Y_{qk}Y_{ql}-\mathbb{E}Y_{qk}Y_{ql})V_{lj}\bigg{|}\geq
t\bigg{)}$ $\displaystyle\leq
2\exp\bigg{(}-c\min\bigg{\\{}\frac{t^{2}}{4\lambda_{1}},\frac{t}{\|M\|}\bigg{\\}}\bigg{)}.$

Setting $t=C\sqrt{\lambda_{1}}\sqrt{\frac{\log(s)+\log(k)+5\log(p)}{n}}$ and
taking a union bound over all $q$ random variables shows that for fixed $i$
and $j$, with probability at least $1-O(s^{-1}k^{-1}p^{-4})$,

$\displaystyle\bigg{|}\bigg{(}\Sigma^{1/2}_{JJ^{c}}\bigg{(}\frac{Y_{J^{c}}^{\top}Y_{J^{c}}}{n}-I\bigg{)}V\bigg{)}_{ij}\bigg{|}$
$\displaystyle\lesssim\sqrt{\lambda_{1}}\sqrt{\frac{\log(p)}{n}}.$

Taking a union bound over $s$ rows and $k$ columns shows that with probability
at least $1-O(p^{-4})$,

$\displaystyle\|\Sigma^{1/2}_{JJ^{c}}\bigg{(}\frac{Y_{J^{c}}^{\top}Y_{J^{c}}}{n}-I\bigg{)}V\|_{\max}$
$\displaystyle\lesssim\sqrt{\lambda_{1}}\sqrt{\frac{\log(p)}{n}}.$

Therefore, from the initial bound in (29) and Proposition 1,

$\displaystyle\|K_{4}\|_{2\to\infty}$
$\displaystyle=\|\Sigma^{1/2}_{JJ^{c}}\bigg{(}\frac{Y_{J^{c}}^{\top}Y_{J^{c}}}{n}-I\bigg{)}VDU_{\perp}^{\top}\tilde{U}_{J}\|_{2\to\infty}$
$\displaystyle\leq\sqrt{s\lambda_{1}}\|\Sigma^{1/2}_{JJ^{c}}\bigg{(}\frac{Y_{J^{c}}^{\top}Y_{J^{c}}}{n}-I\bigg{)}V\|_{\max}\|U_{\perp}^{\top}\tilde{U}_{J}\|$
$\displaystyle\lesssim\lambda_{1}\sqrt{\frac{s\log(p)}{n}}\|U_{\perp}^{\top}\tilde{U}_{J}\|$
$\displaystyle\lesssim\frac{s\log(p)}{n}\frac{\lambda_{1}^{2}}{\lambda_{k}-\lambda_{k+1}}$

as desired. ∎

##  Appendix C Background Material on Orlicz Norms, Concentration, and
Subspace Perturbation

Here we briefly discuss Orlicz $\psi_{\alpha}$ Norms and Bernstein’s
inequality for subexponential random variables.

The _Orlicz Norm_ of order $\alpha$ for a real-valued random variable $X$ is
defined via

$\displaystyle\|X\|_{\psi_{\alpha}}:=\inf\\{t>0:\mathbb{E}\exp(|X|^{\alpha}/t)\leq
1\\}.$

Random variables with finite $\psi_{2}$ norm are called _subgaussian_ and
those with a finite $\psi_{1}$ norm are called _subexponential_. Generally
speaking, if $X$ is subgaussian, then $X^{2}$ is subexponential and
$\|X^{2}\|_{\psi_{1}}\lesssim\|X\|_{\psi_{2}}^{2}$. One also has the “Cauchy-
Schwarz” bound $\|XY\|_{\psi_{1}}\lesssim\|X\|_{\psi_{2}}\|Y\|_{\psi_{2}}$
((Vershynin, 2018).

For subexponential random variables, one has the following generalized
Bernstein’s inequality. See Theorem 2.8.2 in Vershynin ((2018) for the proof.

######  Theorem 3 (Theorem 2.8.2 in Vershynin ((2018)).

Let $X_{1},...,X_{N}$ be independent, mean zero subexponential random
variables and let $a=(a_{i})_{i=1}^{N}$. Then there exists a universal
constant $c>0$ such that for all $t\geq 0$, we have that

$\displaystyle\mathbb{P}\left\\{\left|\sum_{i=1}^{N}a_{i}X_{i}\right|\geq
t\right\\}\leq
2\exp\left[-c\min\left(\frac{t^{2}}{K^{2}\|a\|_{2}^{2}},\frac{t}{K\|a\|_{\infty}}\right)\right]$

where $K=\max_{i}\|X_{i}\|_{\psi_{1}}$.

We also make use of the Hanson-Wright Inequality. See Theorem 6.2.1 in
Vershynin ((2018) for the proof.

######  Theorem 4 (Hanson-Wright Inequality –Theorem 6.2.1 in Vershynin
((2018)).

Let $X_{1},\dots,X_{N}$ be independent, mean-zero subgaussian random
variables. Let $M$ be some fixed $N\times N$ matrix. Then there exists a
universal constant $c>0$ such that for all $t\geq 0$, we have that

$\displaystyle\mathbb{P}\bigg{\\{}\bigg{|}\sum_{k,l}M_{kl}X_{k}X_{l}-\mathbb{E}M_{kl}X_{k}X_{l}\bigg{|}\geq
t\bigg{\\}}$ $\displaystyle\leq
2\exp\bigg{(}-c\min\bigg{\\{}\frac{t^{2}}{K^{4}\|M\|_{F}^{2}},\frac{t}{K^{2}\|M\|}\bigg{\\}}\bigg{)},$

where $K=\max_{i}\|X_{i}\|_{\psi_{2}}$.

We also use several notions from subspace perturbation theory. Suppose $U$ and
$\hat{U}$ are two $d_{1}\times d_{2}$ matrices with orthonormal columns with
$d_{2}\leq d_{1}$. The $\sin\Theta$ distance between the subspaces spanned by
$U$ and $\hat{U}$ is defined as follows. Let
$I-UU^{\top}=U_{\perp}U_{\perp}^{\top}$. Then the (spectral) $\sin\Theta$
distance is defined as

$\displaystyle\|\sin\Theta(U_{1},U_{2})\|:$
$\displaystyle=\|\hat{U}^{\top}U_{\perp}\|.$

Throughout the supplementary material, we use several equivalent terms for the
$\sin\Theta$ distance. We present this here as a lemma, the statement of which
is slightly modified from Lemma 1 of Cai and Zhang ((2018).

######  Lemma 10 (Modified from Lemma 1 of Cai and Zhang ((2018)).

The $\sin\Theta$ distance between two matrices satisfies

$\displaystyle\|\sin\Theta(\hat{U},U)\|\leq\inf_{W:WW^{\top}=I_{d_{2}}}\|\hat{U}-UW\|\leq\sqrt{2}\|\sin\Theta(\hat{U},U)\|;$
$\displaystyle\|\sin\Theta(\hat{U},U)\|\leq\|\hat{U}\hat{U}^{\top}-UU^{\top}\|\leq
2\|\sin\Theta(\hat{U},U)\|.$


#  On Statistical Distribution for Adiabatically Isolated Body

Natalia Gorobey  Peter the Great Saint Petersburg Polytechnic University,
Polytekhnicheskaya 29, 195251, St. Petersburg, Russia  Alexander Lukyanenko
<EMAIL_ADDRESS>Peter the Great Saint Petersburg Polytechnic University,
Polytekhnicheskaya 29, 195251, St. Petersburg, Russia  A. V. Goltsev  Ioffe
Physical- Technical Institute, Polytekhnicheskaya 26, 195251, St. Petersburg,
Russia

###### Abstract

The statistical distribution for the case of an adiabatically isolated body
was obtained in the framework of covariant quantum theory and Wick’s rotation
in the complex time plane. The covariant formulation of the mechanics of an
isolated system lies in the rejection of absolute time and the introduction of
proper time as an independent dynamic variable. The equation of motion of
proper time is the law of conservation of energy. In this case, the energy of
an isolated system is an external parameter for the modified distribution
instead of temperature.

##  I INTRODUCTION

Statistical Boltzmann distribution with density

$f\left(q,p\right)=A\exp\left[-\frac{H\left(q,p\right)}{k_{B}T}\right]$ (1)

for an adiabatically isolated body has an approximate character Landau . It
can be used to calculate the statistical characteristics of small subsystems
of a macroscopic body, but when calculating such quantities as the total
energy and its fluctuations, it leads to obviously incorrect results. Although
the calculation errors in this case are not large compared to the quantities
themselves and may not be taken into account in the macroscopic approximation,
the search for an adequate replacement of distribution Eq.(1) for an
adiabatically isolated body is of fundamental importance.

In this search, we use the formulation of the laws of statistics in quantum
mechanics Fey ,FeyHibbs  and the transparent analogy with quantum dynamics
itself available here. It lies in the analogy between the parabolic equation
for the density matrix,

$\frac{\partial\widehat{\rho}}{\partial\beta}=-\widehat{H}\widehat{\rho},$ (2)

where $\beta=1/k_{B}T$, and the Schrödinger equation for the evolution
operator,

$i\hbar\frac{\partial\widehat{U}}{\partial t}=\widehat{H}\widehat{U}.$ (3)

It can be seen that the evolution operator

$\widehat{U}=\exp\left(\frac{1}{i\hbar}\widehat{H}t\right)$ (4)

turns into a density operator (matrix) by a simple replacement

$t\rightarrow i\hbar\beta,$ (5)

i.e., as a result of the transition to imaginary time. In turn, the classical
Boltzmann distribution Eq. (1) can be obtained from the mixed coordinate-
momentum representation of the logarithm of the density operator due to the
relation

$\beta H\left(q,p\right)=\beta\left\langle
p\right|\widehat{H}\left|q\right\rangle=-\left\langle
p\right|\ln\widehat{\rho}\left|q\right\rangle,$ (6)

which is valid if we agree to place the momentum operators in $\widehat{H}$ to
the right of the coordinates. In FSS , the definition of the partition
function of an adiabatically isolated body in the form of a trace of the
density operator in the modified quantum theory was proposed. The modification
consists in applying the rules of covariant quantization Frad  to a dynamical
system with reparametrization time invariance. In such a system, proper time
is naturally introduced as an additional dynamic variable. Its equation of
motion is the law of conservation of energy of the system. In the transition
to statistical mechanics, as a result of replacement Eq. (5), we obtain a
modified partition function as a function of the energy of an isolated system.
In this paper, we will go further in the modification of statistical
mechanics, and using relation Eq. (6), we will determine the analogue of
distribution Eq. (1) for an isolated body.

The first section formulates the dynamics of an isolated mechanical system in
terms of proper time. The second section proposes a quantum version of the
modified dynamics and a modified statistical distribution.

##  II MECHANICS OF AN ISOLATED BODY

Mechanical laws can be formulated from the outset in a form that explicitly
takes into account the complete isolation of the system under consideration
from its environment. First of all, it concerns time. Instead of the absolute
Newton time $t$, we introduce an arbitrary parameter $\tau\in\left[0,1\right]$
and a new dynamic variable $s\left(\tau\right)$:

$s\left(\tau\right)=\frac{dt}{d\tau}.$ (7)

Accordingly, we write the action integral in the form

$I\left[q,\overset{\cdot}{q},\overset{\cdot}{s}\right]=\int_{0}^{1}d\tau\overset{\cdot}{s}L\left(q,\frac{\overset{\cdot}{q}}{\overset{\cdot}{s}}\right).$
(8)

Obviously, action Eq. (8) is invariant under an arbitrary change of the
parameter $\tau$. The new dynamic variable $s\left(\tau\right)$ will play the
role of the proper time of the system, if we take into account its equation of
motion,

$\frac{\delta I}{\delta s}=\frac{dW}{d\tau}=0,$ (9)

where

$W\equiv\frac{\partial L\left(q,q^{\prime}\right)}{\partial
q_{k}^{\prime}}q_{k}^{\prime}-L,$ (10)

is the energy, $k=1,2,\ldots,K,K$ \- number of degrees of freedom of the
system. Here the prime denotes the derivative with respect to proper time.
Indeed, as we know, the first integral of motion

$W=E$ (11)

allows you to determine your own time by the movement of dynamic variables
$q_{k}\left(s\right)$. It remains only to fix the surface of constant energy
Eq. (11) as an additional condition in the action integral:

$\widetilde{L}\left(q,q^{\prime}\right)=L\left(q,q^{\prime}\right)+\lambda^{\prime}W.$
(12)

We have discarded the total derivative $\lambda^{\prime}E$ in Eq. (12). Note
that the arbitrariness in the choice of the parameter $\tau$ is now expressed
in the arbitrariness of the interval of motion in proper time
$s\in\left[0,C\right]$, which is removed only by the additional condition Eq.
(11).

As a next step, we find the Hamilton function of the modified system. First,
we define the canonical momenta:

$\widetilde{p}_{k}=m_{kl}q_{l}^{\prime}\left(1-\lambda^{\prime}\right),$ (13)

$\widetilde{p}_{\lambda}=W=\frac{1}{2}m_{kl}q_{k}^{\prime}q_{l}^{\prime}+V\left(q\right),$
(14)

where the stroke denotes the derivative with respect to proper time and we
specified the form of the original Lagrange function:

$L\left(q,\overset{\cdot}{q}\right)=\frac{1}{2}m_{kl}\overset{\cdot}{q}_{k}\overset{\cdot}{q}_{l}-V\left(q\right).$
(15)

With the help of these equations we find speeds. At first,

$q_{k}^{\prime}=\frac{m_{kl}^{-1}\widetilde{p}_{l}}{\left(1-\lambda^{\prime}\right)},$
(16)

and then,

$\lambda^{\prime}=1-\frac{\sqrt{m_{kl}^{-1}\widetilde{p}_{k}\widetilde{p}_{l}}}{\sqrt{2\left(E-V\right)}},$
(17)

where Eq. (11) is taken into account. After that, we find the modified
Hamilton function:

$\widetilde{H}=\sqrt{2\left(E-V\right)}\sqrt{m_{kl}^{-1}\widetilde{p}_{k}\widetilde{p}_{l}}+E-2V.$
(18)

Thus, in modified mechanics, the proper time of motion of an isolated body is
an indeterminate parameter (before solving the classical equations of motion),
and the energy has a fixed value. The two-valued root in Eq. (18) corresponds
to two half-cycles of the system’s motion between two cusps in the region of
allowed classical motion. After quantization, a root from the elliptic
operator will arise, which will need to be given meaning.

In the final step, after quantization, we return to absolute time by simply
setting $s=t$. This is acceptable, since the dynamics of an isolated body can
itself serve to measure the time $t$. There is no obstacle to this in the new
dynamics. Indeed, setting $\widetilde{p}_{k}=p_{k}$, from Eq. (13) we obtain
$\lambda^{\prime}=0$. Then relation Eq. (17) is equivalent to Eq. (11), and
$\tilde{H}=E$, as expected for an isolated system with energy $E$. In what
follows, we will omit the ”tilde” in the designation of canonical momenta.

##  III STATISTICAL DISTRIBUTION FOR AN ISOLATED BODY

For convenience, we set here $\hbar=k_{B}=1$. Let us represent the kernel of
the evolution operator of the modified quantum theory as a functional integral
on the phase space FadSlav :

$\displaystyle D$ $\displaystyle\equiv$ $\displaystyle\left\langle
q^{\prime\prime},C\right|\left.q^{\prime},0\right\rangle=\int\prod\limits_{s}\frac{d^{K}pd^{K}q}{2\pi}\exp\left\\{i\int_{0}^{C}ds\left[p_{k}\overset{\cdot}{q}_{k}\right.\right.$
(19)
$\displaystyle-\left.\left.\widetilde{H}\left(q,p,E\right)\right]\right\\}.$

The formal nature of the functional integral in Eq. (19) is aggravated by the
presence of a root from the quadratic form of the momenta in the Hamiltonian.
We make the kernel more definite by turning the functional integral Eq. (19)
into Gaussian over canonical momenta by introducing integrals over additional
variables:

$\displaystyle D$ $\displaystyle=$
$\displaystyle\int\prod\limits_{s}\frac{d^{K}pd^{K}q}{\left(2\pi\right)^{K}}\int_{-\infty}^{+\infty}\prod\limits_{s}\frac{d\chi}{2\pi}\int_{0}^{+\infty}\prod\limits_{s}dQ^{2}$
(20)
$\displaystyle\times\exp\left\\{i\int_{0}^{C}ds\left[p_{k}\overset{\cdot}{q}_{k}+E-2V\right.\right.$
$\displaystyle\left.\left.-Q+\chi\left(2\left(E-V\right)m_{kl}^{-1}\widetilde{p}_{k}\widetilde{p}_{l}-Q^{2}\right)\right]\right\\}.$

It is easy to see that integration first over the variable $\chi$ and then
over the variable $Q^{2}$ leads to the original integral Eq. (19). However,
now integration over canonical momenta can be done in the first place, and
this will be the definition of the square root of the kinetic energy in the
modified quantum theory. In order to avoid cumbersome expressions, we agree in
what follows to omit numerical factors of type $2\pi$. As a result, we get

$\displaystyle D$ $\displaystyle=$
$\displaystyle\int\prod\limits_{s}d^{K}q\int_{-\infty}^{+\infty}\prod\limits_{s}\frac{d\chi}{\left[2i\chi\left(E-V\right)\right]^{K/2}}$
(21)
$\displaystyle\times\int_{-\infty}^{+\infty}\prod\limits_{s}QdQ\exp\left\\{i\int_{0}^{C}ds\left[-Q\right.\right.$
$\displaystyle-\left.\left.\chi
Q^{2}-\frac{1}{4\chi}\frac{m_{kl}q_{k}^{\prime}q_{l}^{\prime}}{2\left(E-V\right)}+E-2V\right]\right\\}..$

To see what the additional integrals in Eq. (21) now lead to, we calculate the
next integral over the variable $Q$ in the semiclassical approximation Fed :

$D\cong\int\prod\limits_{s}\frac{d^{K}q}{\sqrt{E-V}}\Delta\left[W-E\right]\exp\left[i\int_{0}^{C}ds\left(E-2V\right)\right].$
(22)

Here

$\displaystyle\Delta[W-E]\equiv\mathrm{Re}\int_{0}^{\infty}\prod\limits_{s}\left(i\eta\right)^{\left(K-1\right)/2}d\eta$
$\displaystyle\times\exp\left[i\int_{0}^{C}ds\frac{\eta}{4}\left(\frac{1}{2}m_{kl}q_{k}^{\prime}q_{l}^{\prime}+V-E\right)\right],$
(23)

where $\eta=1/\chi(E-V)$ is the new integration variable. As expected,
functional Eq. (23) limits integral Eq. (22) to trajectories that lie entirely
on the surface of constant energy $E$. For such trajectories, the exponent
under the integral sign in Eq. (22) exactly coincides with the initial action
of the system.

Thus, the presence of the square root of the quadratic pulse shape in the
modified Hamilton function Eq. (18) is the source of the generalized $\delta$
-function Eq. (23) in the modified propagator Eq. (22). This is just what is
needed to solve another problem of covariant quantum theory, the problem of
time $C$. A similar problem of determining the propagator of a relativistic
particle within the framework of the rules of covariant quantum theory Frad
is completed by additional integration over proper time within $0\leq
C<\infty$ with the trivial measure $\mu=1$ Gov . In our problem, the proper
time on an arbitrary trajectory $q(s)$ is determined two-valuedly by two
directions of motion, so we should integrate along the full axis
$-\infty<C<\infty$. It should be borne in mind that the time of movement on
the trajectory $q(s)$ is a functional of the trajectory: $C=C[q(s)]$.
Therefore, an additional integral over proper time in the propagator should be
placed before the final summation over all admissible trajectories. Now the
purpose of the proposed modification of mechanics is clear: the generalized
$\delta$-function Eq. (23) will allow removing additional integration over
proper time in covariant quantum theory and giving the propagator a dynamic
meaning, and after the transition to imaginary proper time, a statistical
meaning to the density matrix. The resulting proper time, and in statistical
mechanics, the reciprocal temperature of an adiabatically isolated body is
random variable in this theory. Thus, we obtain the following representation
of the density matrix of an adiabatically isolated body:

$\displaystyle\widetilde{\rho}\left(q^{\prime\prime},q^{\prime}\right)$
$\displaystyle=$ $\displaystyle\int\prod\limits_{s}\frac{d^{K}q}{\sqrt{V-E}}$
(24)
$\displaystyle\times\int_{-\infty}^{+\infty}dC\Delta\left[V-\frac{1}{2}m_{kl}q_{k}^{\prime}q_{l}^{\prime}-E\right]$
$\displaystyle\times\exp\left[-\int_{0}^{C}ds\left(E-2V\right)\right].$

From the quantum mechanical density matrix Eq. (24), we proceed to the
modification of the classical Boltzmann distribution Eq. (1) for an
adiabatically isolated body using relation Eq. (6):

$\widetilde{f}\left(q,p,E\right)=A\exp\left[B\left(q,p,E\right)\right],$ (25)

where

$B\left(q,p,E\right)=\left\langle
q\right|\ln\widetilde{\rho}\left|p\right\rangle.$ (26)

##  IV CONCLUSIONS

Thus, the statistical distribution for an adiabatically isolated body is
defined within the framework of the formalism of reparameterization invariant
mechanics and the corresponding covariant quantization procedure. The
indeterminate proper time parameter arising in this formalism also plays the
role of an indefinite body temperature in modified statistical mechanics. This
is how it should be in a system that is devoid of contact with the environment
and whose energy is fixed. It must also be assumed that the nominal
distribution Eq. (25) in accordance with the ergodic hypothesis Arn  describes
the relative time that the system spends in the vicinity of the point
$\left(q_{k},p_{k}\right)$ of the phase space when moving on the surface of
constant energy $E$.

Distribution Eq. (25) can only be used for a macroscopic body with a large
amount of mechanical energy. In this case, the continuous law of conservation
of mechanical energy Eq. (9) is obviously substantiated in quantum theory in
the semiclassical approximation. At the same time, the modification of the
classical statistical distribution proposed here is justified.

The next step for studying the statistics of an isolated system is the
calculation of the distribution density Eq. (25) for the simplest system - an
ensemble of harmonic oscillators. This will be the subject of subsequent
works.

##  V ACKNOWLEDGEMENTS

We are thanks V.A. Franke for useful discussions.

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