# Dynamic fluctuation-dissipation theory for Generalized Langevin Equations:
constructive constraints, stability and realizability

Massimiliano Giona, Giuseppe Procopio and Chiara Pezzotti
<EMAIL_ADDRESS>Dipartimento di Ingegneria Chimica, Materiali,
Ambiente La Sapienza Università di Roma
Via Eudossiana 18, 00184 Roma, Italy

###### Abstract

Using the initial-value formulation, a dynamic theory for systems evolving
according to a Generalized Langevin Equation is developed, providing more
restrictive conditions on the existence of equilibrium behavior and its
fluctuation-dissipation implications. For systems fulfilling the property of
local realizability, that for all the practical purposes corresponds to the
postulate of the existence of a Markovian embedding, physical constraints,
expressed in the form of dissipative stability and stochastic realizability
are derived. If these two properties are met, Kubo theory is constructively
recovered, while if one of these conditions is violated a thermodynamic
equilibrium behavior does not exist (and this occurs also for “ well-behaved
dissipative systems” according to the classical Kubo theory), with significant
implications in the linear response theory.

Introduction -  The fluctuation-dissipation (FD) theorems developed by Kubo
for Generalized Langevin Equations (GLE) kubo1 ; kubo2  constitute a
cornerstone in statistical physics admitting a huge variety of applications in
all the branches of physics, microhydrodynamics and transport theory,
electrochemistry and dieletric response, abstract linear response theory, etc.
hydro1 ; gen1 ; gen2 , and cellular biology cell1 ; cell2 , involving thermal
fluctuations.

In its original formulation kubo1 , the Kubo theory addresses the
generalization of the equation of motion of a spherical Brownian particle of
mass $m$ and velocity $v(t)$ in a fluid at thermal equilibrium (constant
temperature $T$), in which the dissipative response of the fluid is not
instantaneous but is described via a memory kernel $h(t)$

$m\,\frac{dv(t)}{dt}=-\int_{-\infty}^{t}h(t-\tau)\,v(\tau)\,d\tau+R(t)$ (1)

where $R(t)$ is the stochastic termal force, such that the resulting velocity
process $v(t)$ is stationary. Another basic assumption for $R(t)$ is the
Langevin condition, originally proposed by P. Langevin in 1908 langevin ,
stating that

$\langle R(t)\,v(\tau)\rangle_{\rm eq}=0\qquad\tau\leq t$ (2)

where $\langle\cdot\rangle_{\rm eq}$ is the expected value with respect to the
probability measure of the thermal fluctuations at equilibrium. Eq. (2)
implies that the stochastic force $R(t)$ at time $t$ is independent of the
previous history of particle velocity. An alternative view to eq. (2) has been
developed in felder  (see also bala  for a critical discussion).

The Kubo FD theory is developed in two steps: the determination of the
velocity autocorrelation function $C_{vv}(t)=\langle v(t+\tau)\,\langle
v(\tau)\rangle_{\rm eq}=\langle v(t)\,v(0)\rangle_{\rm eq}$ (due to
stationarity), and this is referred to as the fluctuation-dissipation theorem
of the first kind (FD1k, for short), and the determination of the
autocorrelation function $C_{RR}(t)=\langle R(t)\,R(0)\rangle$ for the thermal
force, referred to as the fluctuation-dissipation theorem of the second kind
(FD2k, for short). The development of the Kubo theory starts from eq. (1) and
involves essentially Fourier analysis, specifically the application of the
Wiener-Khinchin theorem for stationary stochastic processes, imposing: (i) the
Langevin condition eq. (2) and (ii) the value of the intensity of velocity
fluctuations at thermal equilibrium,

$\langle v^{2}\rangle_{\rm eq}=\frac{k_{B}\,T}{m}$ (3)

where $k_{B}$ is the Boltzmann constant. We prefer to use the wording
“properties” instead of ”theorems”, as FD1k and FD2k are two “properties”
based upon specific physical assumptions (valid for some systems, but that can
be equally well be violated by others), rather that propositions involving
mathematical entities.

It is possible to frame the same problem in a slightly different way,
replacing eq. (1) with

$m\,\frac{dv(t)}{dt}=-\int_{0}^{t}h(t-\tau)\,v(\tau)\,d\tau+R(t)$ (4)

equipped with the initial condition $v(0)=v_{0}$. The comparison of these two
equations shows that the only difference between them refers to the lower
integration value, that is $\tau=-\infty$ in eq. (1) that is turned to
$\tau=0$ in eq. (4). Probably because of the apparent “tininess” of this
difference, no great attention has been focused on it, and in the literature
these two formulations are used interchangeably depending upon practical
convenience lit1 ; lit2 ; lit3 ; lit4 . Henceforth we refer to eq. (1) as the
abstract formulation of the GLE and to eq. (4) as its initial-value
formulation.

In point of fact, the mathematical structure of these two formulations is
different (although intrinsic analogies exist jap ; jap1 ), and more
importantly this difference admits relevant implications as regards the
methodological way of developing FD theory and the results following from it.
The abstract formulation is essentially focused at determining the statistical
properties of the stochastic processes $v(t)$ and $R(t)$ at equilibrium. In
this way, it fits the purposes of an equilibrium theory for the linear
response of physical systems to perturbations (Kramers-Krönig relations,
analysis of susceptibilities, etc.) kubo2 ; degroot . As it represents
intrinsically an equilibrium formulation, it is unable to handle neither the
momentum relaxation dynamics and the convergence towards mechanical
equilibrium nor any experiments in which a particle, possessing initially a
velocity $v_{0}$, is injected into the fluid at some initial time, say $t=0$.
Conversely, the initial-value formulation is a dynamic, non-equilibrium (as
regards momentum transfer, not thermal effects) description of the process,
i.e. of the hydromechanical interactions between the particle and the fluid.
It corresponds to the description of experiments involving a particle (or a
system of independent particles), injected into the fluid and relaxing its
momentum dynamics towards equilibrium conditions, providing a stochastic
evolution equation for the particle velocity, the results of which can be
directly checked against experiments such as those involving Brownian motion
exp0 ; exp1 ; exp2 ; exp3 . For the sake of physical correctness, eq. (4)
describes the fluid-particle interactions in a viscous fluid neglecting fluid-
inertial effects franosch ; procopiovisco , in which the memory kernel $h(t)$
accounts for the viscoelastic dissipative properties of the fluid rheol1 ;
rheol2 .

The Kubo FD theory has been developed starting from eq. (1) and there is no
direct proof of it starting from eq. (4). The present Letter is aimed at
developing the necessary technical tools for a FD theory starting from eq.
(4), and derive the physical implications. This change of perspective permits
to derive new and apparently unexpected properties of FD theory, associated
with its physical realizability, existence of equilibrium properties, and
ultimately applicability of the Stokes-Einstein relation for simple and “well-
behaved” dissipative memory kernels. The concept of local realizability of the
memory kernel, dissipative stability of the GLE, and stochastic realizability
for the thermal fluctuations are introduced as necessary constraints
corresponding to thermodynamic consistency conditions. Depending on their
fulfillment several different regimes are observed and explained. The results
obtained are not only consistent with the mathematical theory of linear
integrodifferential equations math1 ; math2 , but also show unexpected
features once compared and contrasted against the classical FD theory: in
apparently “simple well-behaved dissipative” systems (see further for a
definition), for which the classical Kubo theory predicts a regular diffusive
behavior satisfying Stokes-Einstein relations, the present theory correctly
predicts and explains the violation of the Stokes-Einstein relation as a
consequence of the lack of any thermodynamic equilibrium behavior. Moreover,
the present theory provides a simple interpretation of the phenomena
associated with ergodicity breaking of simple dissipative GLE in the presence
of memory kernels exponentially decaying in time bao1 ; bao2 ; bao3 ;
plyukhin1 ; plyukhin2 . For a thorough analysis of this case see ergbreak .
The present approach constitutes a necessary complement and a significant
refinement of the linear response theory as regards thermodynamic consistency
of the memory kernels and of their interaction with equilibrium fluctuations.

FD1k and FD2k - To begin with, consider FD1k. This follows directly from eq.
(4) as a consequence of the Langevin condition eq. (2), multiplying it by
$v(0)$ and taking the expected value with respect to the equilibrium
probability measure:

$m\,\frac{d\langle v(t)\,v(0)\rangle_{\rm
eq}}{dt}=m\,\frac{dC_{vv}(t)}{dt}=-\int_{0}^{t}h(t-\tau)\,C_{vv}(\tau)\,d\tau$
(5)

equipped with the initial condition $C_{vv}(0)=\langle v^{2}\rangle_{\rm eq}$.
In the Laplace domain, setting
$\widehat{h}(s)=L[h(t)]=\int_{0}^{\infty}e^{-st}\,h(t)\,dt$, and similarly for
the other functions, the solution of eq. (5) is

$\widehat{C}_{vv}(s)=\frac{\langle v^{2}\rangle}{s+\widehat{h}(s)/m}=\langle
v^{2}\rangle\,\widehat{G}(s)$ (6)

where $\widehat{G}(s)$ is the Laplace transform of the Green function
(referred to as the resolvent in math1  associated with the memory dynamics
defined by $h(t)$).

Elaborating further the structure of the stochastic perturbation (see the
Appendices), it is possible to derive the following condition

$\int_{0}^{\infty}G(\tau)\,\left[k_{B}\,T\,h(\tau)-C_{RR}(\tau)\right]\,d\tau=0$
(7)

where $G(t)=L^{-1}[\widehat{G}(s)]$ is the resolvent function defined by eq.
(6). Eq. (7) is the most general FD2k result associated with the analysis of
the kinetic energy for the initial value GLE. It will be referred to as the
weak formulation of FD2k. Of course, if

$C_{RR}(t)=k_{B}\,T\,h(t)$ (8)

corresponding to the Kubo FD2k for abstract GLE, eq. (7) is identically
satisfied, but we cannot claim eq. (8) directly from eq. (7).

Local realizability -  In order to improve the analysis, a further condition
on the GLE should be posed. Indeed, a solution to this problem is provided by
the assumption of local realizability, introduced below. Consider again the
memory term $-h(t)*v(t)$ in eq. (4) where “$*$” stands for convolution,
corresponding to the force exerted by the fluid on the particle at time $t$,
and a continuous kernel $h(t)$ (i.e. not impulsive as in the case of the
Stokes-Einstein kernel, $h(t)=\eta\,\delta(t)$). This action is local in time,
in the meaning that it should be properly viewed as the lumped and compact
mathematical representation of local effects, occurring instantaneously in
time, i.e. at time $t$. This means that a vector-valued $n$-dimensional
function ${\bf z}(t)$ of time and a scalar function
$\psi:{\mathbb{R}}^{n}\rightarrow{\mathbb{R}}$ should exist, such that
$-h(t)*v(t)=\psi({\bf z}(t))$, where ${\bf z}(t)$ evolves according to a
differential equation $d{\bf z}(t)/dt={\bf f}({\bf z}(t),v(t),t)$.

But eq. (4) is linear and it gives rise to a stationary stochastic process
$v(t)$. This implies that the vector field ${\bf f}({\bf z}(t),v(t),t)$ should
be linear in both ${\bf z}(t)$ and $v(t)$ and autonomous (time-independent),
and $\phi({\bf z}(t))$ a linear (and homogeneous) functional of ${\bf z}(t)$.
Consequently, the memory term entering eq. (4) should be the expressed as the
linear projection of the local functions ${\bf
z}(t)=(z_{1}(t),\dots,z_{n}(t))$, be their system finite or countable, where
${\bf z}(t)$ satisfies an ordinary linear differential equation with constant
coefficients driven by stochastic fluctuations, that can be expressed as a
linear combination of white noise processes (by eq. (2)).

We thus arrive at the definition of local realizability. The GLE is said to be
locally realizable if there exist a constant $n\times n$ matrix $\bm{\Lambda}$
and a constant $n$-vector ${\bf a}$ such that eq. (4) for $R(t)=0$ can be
expressed as the projection with respect to ${\bf a}$ of an $n$-dimensional
process of internal degrees of freedom ${\bf z}(t)$,

$m\,\frac{dv(t)}{dt}=-\sum_{h=1}^{n}a_{i}\,z_{i}(t)=-\left({\bf a},{\bf
z}(t)\right)$ (9)

evolving according to the local dynamics

$\frac{d{\bf z}(t)}{dt}=-\bm{\Lambda}\,{\bf z}(t)+{\bf 1}\,v(t)$ (10)

where ${\bf 1}=(1,\dots,1)$. From the mathematical point of view, local
realizability is one-to-one with the property that $h(t)$ possesses a finite
or countable number of pole singularities. It follows from this definition
that

$h(t)=\left({\bf a},e^{-\bm{\Lambda}\,t}\,{\bf 1}\right)$ (11)

The GLE is stocastically realizable if there exists a constant $n\times n$
matrix $\bm{\beta}$ such that eq. (4) can be expressed by eq. (9) with

$\frac{d{\bf z}(t)}{dt}=-\bm{\Lambda}\,{\bf z}(t)+{\bf
1}\,v(t)+\sqrt{2}\,\bm{\beta}\,\bm{\xi}(t)$ (12)

where $\bm{\xi}(t)=(\xi_{1}(t),\dots,\xi_{n}(t))$ is a vector-valued white
noise process vankampen ,
$\langle\xi_{i}(t)\,\xi_{j}(\tau)\rangle=\delta_{ij}\,\delta(t-\tau)$, and
eqs. (9), (12) admit an equilibrium invariant measure for which eqs. (3) and
(5) hold (i.e. FD1k is verified). In practice, one may choose
$\xi_{i}(t)=dw_{i}(t)/dt$, where $w_{i}(t)$ are independent Wiener processes.

It is easy to observe that local realizability involves exclusively the
dissipative contribution, while stochastic realizability corresponds to the
existence of a Markov embedding of eq. (4), and specifically the existence of
a stochastic process $R(t)$, with the property of fulfilling the Langevin
condition eq. (2) and in turn eqs. (3), (5).

Without loss of generality, assume that the internal degrees of freedom
$z_{i}(t)$ are expressed in the eigenbasis of $\bm{\Lambda}$ so that
$\bm{\Lambda}=\mbox{diag}(\lambda_{1},\dots,\lambda_{n})$. Assume
$\lambda_{i}\in{\mathbb{R}}$, with $\lambda_{i}>0$, that corresponds to the
dissipative friction factor in viscoelastic fluids rheol1 ; rheol2 . At the
moment consider the case where $a_{i}>0$, $i=1,\dots,n$, as dictated by the
rheological analysis rheol1 ; rheol2 . In this case, eq. (10) takes the
simpler form $h(t)=\sum_{i=1}^{n}a_{i}\,e^{-\lambda_{i}\,t}$.

To begin with, assume that the stochastic fluctuations acting on the internal
models are characterized by the commutative property
$[\bm{\Lambda},\bm{\beta}]=\bm{\Lambda}\bm{\beta}-\bm{\beta}\bm{\Lambda}=0$,
and consequently the matrix $\bm{\beta}$ is also diagonal once expressed in
the eigenbasis of $\bm{\Lambda}$, $\bm{\beta}=\mbox{diag}(c_{1},\dots,c_{n})$.

Under these hypotheses, eqs (8), (12) admit an equilibrium invariant measure,
and FD1k, i.e. eq. (5), is fulfilled provided that the conditions

$\langle z_{i}\,v\rangle_{\rm eq}=0\,,\qquad i=1,\dots,n$ (13)

are met (see the Appendices), indicating that the internal variables
$z_{i}(t)$ should be uncorrelated from the velocity $v(t)$ at equilibrium.
Enforcing eq. (3), the expression for the coefficients $c_{i}$ follows

$c_{i}^{2}=m\,\langle v^{2}\rangle\,\frac{\lambda_{i}}{a_{i}}\,,\qquad
i=1,\dots,n$ (14)

consistently with the analysis developed by Goychuk goychuk . Since the
process $R(t)$ attains at equilibrium the expression
$R(t)=-\sqrt{2}\,\sum_{i=1}^{n}a_{i}\,c_{i}\,e^{-\lambda_{i}\,t}*\xi_{i}(t)$,
from eq. (12) it readily follows that (see the Appendices)

$C_{RR}(t)=k_{B}\,T\,\sum_{i=1}^{n}a_{i}e^{-\lambda_{i}\,t}=k_{B}\,T\,h(t)$
(15)

i.e. that FD2k in its strong form is verified.

Consistency conditions: Thermodynamic equilibria and FD failures -  Before
considering the case of negative $a_{i}^{\prime}s$, let us address a
preliminary observation. FD theory applies if a thermodynamic equilibrium
state exists, which is a property that cannot be given for granted for GLE,
even involving “well-behaved” dissipative kernels. The latter property can be
defined by the conditions: (i) $h(t)\geq 0$; (ii) $h(t)\in L_{1}([0,\infty))$,
and thus $0<\eta_{\infty}=\int_{0}^{\infty}h(t)dt<\infty$; (iii) $h(t)$ is
exponentially decaying with time $t$ (for large $t$). In the classical FD
theory property (i)+(ii) ensures that the Laplace-Fourier transform of $h(t)$
exists, and that the effective diffusivity $D$ fulfils the generalized Stokes-
Einstein relation $D=\widehat{C}_{vv}(0)=k_{B}\,T/\eta_{\infty}$. But this is
not true in general. Let us consider a simple example and derive out of it
some general conclusions. Consider a non-dimensional formulation of the GLE
which implies that we can set in eq. (4) $m=1$, and $\langle v^{2}\rangle_{\rm
eq}=1$, and a simple two-mode kernel

$h(t)=a_{1}\,e^{-\lambda_{1}t}+a_{2}\,e^{-\lambda_{2}t}$ (16)

with $a_{1}=1$, $a_{2}=-\alpha$, $\lambda_{1}=1/10$, $\lambda_{2}=1$, where
$\alpha\leq 1$ is taken as a parameter. The graphs of $h(t)$ for several
values of $\alpha$ are depicted in figure 4.

Figure 1: $h(t)/h(0)$ vs $t$ for the model considered in the main text. The
arrow indicates increasing values of $\alpha=-0.1,\,0,\,0.1,\,0.3$. Line (a)
refers to $\alpha=0.1$, line (b) to $\alpha=0.3$.

For these kernels, the conditions (i)-(iii) stated above are met and
$\eta_{\infty}=a_{1}/\lambda_{1}+a_{2}/\lambda_{2}=10-\alpha$. Setting ${\bf
y}=(v,z_{1},z_{2})$ the overall dynamics eqs. (9)-(12) can be compactly
expressed as $d{\bf y}/dt={\bf A}\,{\bf y}+\sqrt{2}\,\bm{\beta}\,\bm{\xi}$,
where the coefficient matrix ${\bf A}$ is given by

${\bf A}=\left(\begin{array}[]{ccc}0&-a_{1}&\alpha\\\ 1&-\lambda_{1}&0\\\
1&0&-\lambda_{2}\end{array}\right)$ (17)

The eigenvalue spectrum of ${\bf A}$ controls the dynamic response of the
system. For $\alpha\in(-\infty,\alpha_{\rm ds})={\mathcal{D}}_{\rm ds}$, with
$\alpha_{\rm ds}=0.21$ all the eigenvalues possess negative real part: for
$\alpha\in(0,\alpha_{\rm ds})$ there exists a real eigenvalue and a couple of
complex conjugate eigenvalues. Above $\alpha_{\rm ds}$, the real part of the
complex-conjugate pair becomes positive, while the real eigenvalue remains
negative. Therefore, for $\alpha>\alpha_{\rm ds}$ the stochastic dynamics is
unstable and the system does not attain an equilibrium behavior, as the
fluctuations cannot be restricted to the stable eigenspace of ${\bf A}$. We
refer to ${\mathcal{D}}_{\rm ds}=(-\infty,\alpha_{\rm ds})$ as to the region
of dissipative stability of the GLE. Outside ${\mathcal{D}}_{\rm ds}$ no
thermodynamic equilibrium exists, because the “dissipative” dynamics of the
GLE is unstable.

This result follows also from the general theory of linear integro-
differential equations math1 ; math2 : the condition for asymptotic stability
of eq. (4) is that $(m\,s+\widehat{h}(s))\neq 0$ for $\mbox{Re}[s]\geq 0$,
corresponding to the absence of eigenvalues of ${\bf A}$ with positive real
part. In this regard, the occurrence of instabilities in GLE with “well-
behaved” kernels goes beyond the assumption of local realizability of $h(t)$.
For GLE with “well-behaved” kernels that are not dissipatively stable, the
Fourier-Laplace transform of $h(t)$ is defined outside the half-plane of
convergence of $\widehat{h}(s)$ (that is $\mbox{Re}[s]>\mu_{\rm max}>0$, with
$\mu_{\rm max}>0$ the largest positive real part of the eigenvalues of ${\bf
A}$) and, as a consequence, it neither describes the response to external
perturbations (for instance periodic ones) nor it permits to derive the
expression for the diffusivity (that in these situations simply does not
exist). As a further remark, the problem of ergodicity breaking considered in
bao1 ; bao2 ; bao3 ; plyukhin1 ; plyukhin2  corresponds to conditions on the
kernel lying at the boundary of the region of dissipative stability ergbreak .

We can use further eq. (16) as a model for unveiling the constructive
structure of the FD theory. From what said above, the region
$\alpha\in{\mathcal{D}}_{\rm c}=(-\infty,0]\subset{\mathcal{D}}_{\rm ds}$, and
consequently an equilibrium state exists. Moreover, we have obtained
constructively that the matrices $\bm{\Lambda}$ and $\bm{\beta}$ commute, and
that the FD theory applies. Let $\alpha_{\rm c}=0$ be the upper value of
$\alpha$ in this region.

For $\alpha\in(\alpha_{\rm c},\alpha_{\rm ds})$ the eigenrepresentation of
$\Lambda$ does not diagonalize $\bm{\beta}$, and therefore $\bm{\beta}$ should
be a full matrix. The mathematical details are developed in the Appendices,
and rely on the properties of $\bm{\beta}$. It is shown that for
$\alpha=(\alpha_{\rm c},\alpha_{\rm nc})$ with $\alpha_{\rm nc}=0.1$, a non-
diagonal matrix $\bm{\beta}$ can always be defined, such that FD1k and FD2k
are verified. Conversely, for $\alpha\in(\alpha_{\rm nc},\alpha_{\rm ds})$
there is no stochastic model consistent with the FD1k eq. (5) although, for
any choice of $\bm{\beta}$, the stochastic dynamics eqs. (9),(12) admits a
stationary non-equilibrium invariant measure. For
$\alpha={\mathcal{D}}_{sr}=(-\infty,\alpha_{nc})$ the GLE is stochastically
realizable and this means that: (i) the GLE is not unstable, and (ii) a model
for the thermal force can be realized such that the corresponding stationary
measures/fluctuations correspond to those in thermal equilibrium conditions.
It follows from (i) that ${\mathcal{D}}_{sr}\subseteq{\mathcal{D}}_{\rm ds}$,
(ii) $\alpha_{\rm c}\leq\alpha_{\rm nc}\leq\alpha_{\rm ds}$, and consequently
the FD theory applies for kernels belonging to ${\mathcal{D}}_{sr}$. To make a
numerical example, figure 5 (upper panel) depicts the velocity autocorrelation
functions for $\alpha=0.08\in{\mathcal{D}}_{\rm sr}$ (inner dots), and the
“ideal” Kubo autocorrelation function $C_{vv}^{(K)}(t)$ solution of eq. (5)
for $\alpha=0.18$ (curve a) outside ${\mathcal{D}}_{\rm sr}$ but still inside
${\mathcal{D}}_{ds}$ that, from what said above, cannot be realized by any
stochastic model consistent with eqs. (3), (5). The lower panel corresponds to
$C_{vv}^{(K)}(t)$ for $\alpha=0.3$, outside ${\mathcal{D}}_{\rm ds}$, for
which $|C_{vv}^{(K)}(t)|\sim e^{\mu_{\rm max}\,t}$ where $\mu_{\rm max}\simeq
0.0214$.

Figure 2: Upper panel: $C_{vv}(t)$ (symbols) for $\alpha=0.08$ and
$C_{vv}^{(K)}(t)$ for $\alpha=0.18$ vs $t$. Lower panel: $C_{vv}^{(K)}(t)$
(line a) for $\alpha=0.3$ vs $t$. Line (b) corresponds to the exponential
scaling $e^{\mu_{\rm max}\,t}$.

The properties of “well-behaved” kernels in the region outside
${\mathcal{D}}_{\rm ds}$ admits relevant implications in the linear response
theory. Specifically, if $h(t)$ possesses real poles with negative real part
this does not imply that its response to a sinusoidal perturbation ${\bf
F}(\omega)=F_{0}e^{\pm\mathrm{i}\omega\,t}$ would be bounded and described by
the mobility function $\mu(\omega)=1/(i\omega+h[\omega]$) kubo2 , where
$h[\omega]=\widehat{h}(\mathrm{i}\,\omega)$, $\mathrm{i}=\sqrt{-1}$.
Generically we have an exponentially diverging dynamics with time $t$, where
$|v(t)|\leq\mu_{c}^{(e)}(\omega)\,F_{0}\,e^{\mu_{\rm max}\,t}$, if
$F(\omega)=F_{0}\,\cos(\omega\,t)$, and
$|v(t)|\leq\mu_{s}^{(e)}(\omega)\,F_{0}\,e^{\mu_{\rm max}\,t}$, if
$F(\omega)=F_{0}\,\sin(\omega\,t)$, with obviously
$\mu_{s}^{(e)}(\omega)\neq\mu_{c}^{(e)}(\omega)$. The graph of these functions
is depicted in figure 6 for $\alpha=0.3$. In these cases, i.e. for
dissipatively unstable systems, the classical linear response theory should be
revisited. It follows straightforwardly that, outside ${\mathcal{D}}_{\rm ds}$
no Stokes-Einstein relations can be defined, albeit
$\eta_{\infty}=\int_{0}^{\infty}h(t)\,dt>0$ may exist.

Figure 3:  $\mu(\omega)$ line (a), $\mu_{c}^{(e)}(\omega)$ line (b), and
$\mu_{s}^{(e)}(\omega)$ line (c) predicted analytically (see the Appendices).
Symbols, are the results of numerical simulations.

Concluding remarks -  The theory above developed provides clear thermodynamic
conditions to be set for the memory response function $h(t)$ of physical
equilibrium systems. In order to be thermodynamically consistent, $h(t)$
should be: dissipatively stable, as otherwise it is meaningless to assume the
occurrence of stationary properties (and a-fortiori the existence of an
equilibrium state), and (ii) stochastically realizable, as otherwise the
steady states do not correspond to thermodynamic equilibrium conditions driven
by stochastic fluctuations. The first property involves exclusive the
dissipative dynamics, i.e. the kernel $h(t)$. The condition of stochastic
realizability provides a constructive and definite limit to the FD theory. The
application of FD theory for systems outside the region ${\mathcal{D}}_{sr}$
is not only mathematically incorrect, but also may lead to completely
erroneous physical predictions. Retrospectively, the “wormhole” of the
classical FD theory based on eq. (1) appears clear: since it assumes the
occurrence of thermodynamic equilibrium conditions without assessing them, it
fails whenever the equilibrium conditions cannot occur either by dissipative
instability or by the lack of stochastic realizability. This shortcoming
cannot occur within the initial-value formulation of FD theory, as it is
grounded on a constructive approach to FD2k. This is the principal
methodological difference in the two approaches, as from the mathematical
point of view the conditions assessing the asymptotic stability of eq. (4) are
fully equivalent to those for eq. (1) jap . The analysis of the dissipative
stability of “well-behaved kernels”, such as those defined by eq. (16),
indicates further that the physical meaning of dissipation is much more subtle
either than the existence of a positive and finite value of $\eta_{\infty}>0$
and $\mbox{Re}[\lambda_{i}]>0$, or than the condition $\mbox{trace}({\bf
A})<0$, as it necessarily involves a description of the internal dynamics in
order to assess its thermodynamic plausibility. Moreover, it is possible to
show the breakdown of the constraints of dissipative stability and stochastic
realizability ever for monotonically non-increasing kernels, i.e. such that
$dh(t)/dt\leq 0$ for all $t\geq 0$. This hinges towards a constructive
approach to FD theory grounded on the explicit representation of $R(t)$ in
terms of stochastics processes, as developed above. A thorough elaboration of
the dynamic theory of GLE will be developed in a longer communication, with
particular emphasys to hydromechancal interactions and Brownian motion and to
the Debye relaxation mechanism debye1 ; debye2 .

Appendix I - Kinetic energy balance - Kinetic energy balances represent a
convenient way to approach FD2k starting from the initial-value
representation. To begin with, consider the classical memoryless Langevin
model for a Brownian spherical particle, where the thermal force is
proportional to the distributional derivative of a Wiener process
$R(t)=\alpha\,dw(t)/dt$,

$m\,dv(t)=-\eta\,v(t)\,dt+\alpha\,dw(t)$ (18)

and $\eta$ is the friction factor. Consider the evolution equation for the
kinetic energy. Due to the singular nature of eq. (18), the Ito lemma can be
applied,

$\displaystyle\frac{m}{2}dv^{2}(t)$ $\displaystyle=$ $\displaystyle
m\,v(t)\,dv(t)+\frac{m}{2}\left(dv(t)\right)^{2}$ (19) $\displaystyle=$
$\displaystyle-\eta\,v^{2}(t)\,dt+\alpha\,v(t)\,dw(t)+\frac{\alpha^{2}}{2\,m}\,dt+o(dt)$

Taking the expected value of both sides of eq. (19) with respect to the
probability measure of the fluctuations at equilibrium, enforcing the Langevin
condition, i.e. $\langle v(t)\,dw(t)\rangle=0$, and considering that at
equilibrium $d\,\langle v^{2}\rangle_{\rm eq}=0$, we obtain to the $dt$-order,
$-\eta\,\langle v^{2}\rangle+\alpha^{2}/2\,m=0$, leading to
$\alpha=\sqrt{2\,k_{B}\,T\,\eta}$.

The same approach can be applied to the GLE. Due to the convolutional nature
of the dissipation term, it is natural to consider a convolutional expression
for $R(t)$, i.e.,

$m\,\frac{dv(t)}{dt}=-\int_{0}^{t}h(t-\tau)\,v(\tau)\,d\tau+\int_{0}^{t}\gamma(t-\tau)\,\xi(\tau)\,d\tau$
(20)

where $\xi(t)$ is the distributional derivative of a Wiener process, and
$\gamma(t)$ a smooth memory kernel. In this case, the stochastic force is
given by $R(t)=\int_{0}^{t}\gamma(t-\tau)\,\xi(\tau)\,d\tau$, and its
autocorrelation function can be expressed as

$C_{RR}(t)=\langle
R(t)\,R(0)\rangle=\int_{0}^{\infty}\gamma(t+\tau)\,\gamma(\tau)\,d\tau$ (21)

Similarly to the impulsive case, one can to derive FD2k by considering the
evolution of the kinetic energy. From eq. (20) we obtain

$\displaystyle\frac{m}{2}\,\frac{d\langle v^{2}(t)\rangle}{dt}$
$\displaystyle=$ $\displaystyle-\int_{0}^{t}h(t-\tau)\,\langle
v(\tau)\,v(t)\rangle\,d\tau+\int_{0}^{t}\gamma(t-\tau)\,\langle\xi(\tau)\,v(t)\rangle\,d\tau$
$\displaystyle=$
$\displaystyle-\int_{0}^{t}h(t-\tau)\,C_{vv}(t-\tau)\,d\tau+\int_{0}^{t}\gamma(t-\tau)\,C_{v\xi}(t-\tau)\,d\tau$
$\displaystyle=$
$\displaystyle-\int_{0}^{t}h(\tau)\,C_{vv}(\tau)\,d\tau+\int_{0}^{t}\gamma(\tau)\,C_{v\xi}(\tau)\,d\tau$

that has been derived by enforcing the property that $v(t)$ is a stationary
stochastic process. At equilibrium, i.e. for $t\rightarrow\infty$, the l.h.s.
of eq. (Dynamic fluctuation-dissipation theory for Generalized Langevin
Equations: constructive constraints, stability and realizability) is
vanishing, and this implies

$\int_{0}^{\infty}h(\tau)\,C_{vv}(\tau)\,d\tau=\int_{0}^{\infty}\gamma(\tau)\,C_{v\xi}(\tau)\,d\tau$
(23)

where $C_{vv}(t)=\langle v(t)\,v(0)\rangle_{\rm eq}$ and $C_{v\xi}(t)=\langle
v(t)\,\xi(0)\rangle_{\rm eq}$ are the correlation functions at equilibrium
conditions. The evolution equation for $C_{v\xi}(t)$ follows from eq. (20),
multiplying it by $\xi(0)$ and taking the expect value

$m\,\frac{dC_{v\xi}(t)}{dt}=-\int_{0}^{t}h(t-\tau)\,C_{v\xi}(\tau)\,d\tau+\gamma(t)$
(24)

It is equipped with the initial condition $C_{v\xi}(0)=0$. Thus,

$\displaystyle C_{vv}(t)$ $\displaystyle=$ $\displaystyle\langle
v^{2}\rangle\,G(t)$ $\displaystyle C_{v\xi}(t)$ $\displaystyle=$
$\displaystyle\int_{0}^{t}G(t-\tau)\,\gamma(\tau)\,d\tau=\frac{1}{m}\,G(t)*\gamma(t)$
(25)

where $G(t)$ is the Green function expressed by the inverse Laplace transform

$G(t)=L^{-1}\left[\frac{1}{s+\widehat{h}(s)/m}\right]$ (26)

Substituting eq. (25) into eq. (23) we have

$m\,\langle
v^{2}\rangle\int_{0}^{\infty}h(\tau)\,G(\tau)\,d\tau=\underbrace{\int_{0}^{\infty}\gamma(\tau)\,\left[\gamma*G(\tau)\right]\,d\tau}_{I}$
(27)

The integral $I$ at the r.h.s of eq. (27) can be expressed as

$\displaystyle I$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\int_{0}^{\tau}\gamma(\tau-\theta)\,G(\theta)\,d\theta=\int_{0}^{\infty}G(\theta)\,d\theta\int_{\theta}^{\infty}\gamma(\tau)\,\gamma(\tau-\theta)\,d\tau$
(28) $\displaystyle=$
$\displaystyle\int_{0}^{\infty}G(\theta)\,\int_{0}^{\infty}\gamma(\tau+\theta)\,\gamma(\tau)\,d\tau=\int_{0}^{\infty}G(\theta)\,C_{RR}(\theta)\,d\theta$

and therefore eq. (27) takes the final form

$\int_{0}^{\infty}G(\tau)\,\left[k_{B}\,T\,h(\tau)-C_{RR}(\tau)\right]\,d\tau=0$
(29)

corresponding to the weak formulation of the FD2k reported in the main text.

Appendix II - Stochastic realizations: the commutative case -  This case
corresponds to the mutual diagonalization of the dissipative and fluctuational
contributions, (i.e. of the matrices $\bm{\Lambda}$ and $\bm{\beta}$). Here we
outline the basic methodology used, while for the explicit calculations we
refer to the next section, dealing with the more general condition
$[\bm{\Lambda},\bm{\beta}]\neq 0$. Consider the expression for the kernel
$h(t)$

$h(t)=\sum_{i=1}^{n}a_{i}\,e^{-\lambda_{i}\,t}$ (30)

with $\lambda_{i}>0$. To this kernel, assuming $[\bm{\Lambda},\bm{\beta}]=0$,
corresponds the local representation of the GLE

$\displaystyle\frac{dv(t)}{dt}$ $\displaystyle=$
$\displaystyle-\sum_{i=1}^{n}a_{i}\,z_{i}(t)$
$\displaystyle\frac{dz_{i}(t)}{dt}$ $\displaystyle=$
$\displaystyle-\lambda_{i}\,z_{i}(t)+v(t)+\sqrt{2}\,c_{i}\,\xi_{i}(t)\,,\quad
i=1,\dots,n$ (31)

where $\xi_{i}(t)$ can be taken as distributional derivatives of independent
Wiener processes $w_{i}(t)$, $\xi_{i}(t)=dw_{i}(t)/dt$, $i=1,\dots,n$. In this
way, the associated Fokker-Planck equation for the probability density
$p(v,{\bf z},t)$, ${\bf z}=(z_{1},\dots,z_{n})$ is parabolic. The moments

$\displaystyle\langle v^{2}(t)\rangle$ $\displaystyle=$ $\displaystyle\int
v^{2}\,p(v,{\bf z},t)\,dvd{\bf z}$ $\displaystyle\langle
z_{i}(t)\,v(t)\rangle$ $\displaystyle=$ $\displaystyle\int z_{i}\,v\,p(v,{\bf
z},t)\,dvd{\bf z}=y_{i}(t)\,,\quad i=1,\dots,n$ (32) $\displaystyle\langle
z_{i}(t)\,z_{j}(t)\rangle$ $\displaystyle=$ $\displaystyle\int
z_{i}\,z_{j}\,p(v,{\bf z},t)\,dvd{\bf z}=m_{i,j}(t)\,,\quad i,j=1,\dots,n$

can be defined, where $d{\bf z}=dz_{1}\cdots dz_{n}$, and their evolution
equations derived, equipped with arbitrary initial conditions associated with
the initial preparation of the process, defined e.g. by any initial density
$p_{0}(v,{\bf z})$, $p(v,{\bf z},t=0)=p_{0}(v,{\bf z})$. At equilibrium, if an
equilibrum exists, the steady-state moment values are given by the equilibrium
quantities $\langle v^{2}\rangle_{\rm eq}$, $\langle z_{i}\,v\rangle_{\rm
eq}$, $\langle z_{i}\,z_{j}\rangle_{\rm eq}$, respectively.

Similarly, by enforcing the Langevin condition, it follows for the correlation
functions $C_{vv}(t)$ and $C_{z_{i}v}(t)$ at equilibrium,

$\displaystyle C_{vv}(t)$ $\displaystyle=$ $\displaystyle\langle
v(t)\,v(0)\rangle_{\rm
eq}=\lim_{\tau\rightarrow\infty}\langle(v(t+\tau)\,v(\tau)\rangle$
$\displaystyle C_{z_{i}v}(t)$ $\displaystyle=$ $\displaystyle\langle
z_{i}(t)\,v(0)\rangle_{\rm
eq}=\lim_{\tau\rightarrow\infty}\langle(z_{i}(t+\tau)\,v(\tau)\rangle\,,\qquad
i=1,\dots,n$ (33)

the system of evolution equations

$\displaystyle\frac{dC_{vv}(t)}{dt}$ $\displaystyle=$
$\displaystyle-\sum_{i=1}^{n}a_{i}\,C_{z_{i}v}(t)$
$\displaystyle\frac{dC_{z_{i}v}(t)}{dt}$ $\displaystyle=$
$\displaystyle-\lambda_{i}\,C_{z_{i}v}(t)+C_{vv}(t)$ (34)

equipped with the initial conditions,

$C_{vv}(0)=\langle v^{2}\rangle_{\rm eq}\,,\qquad C_{z_{i}v}(0)=\langle
z_{i}\,v\rangle_{\rm eq}\,,\quad i=1,\dots,n$ (35)

From the second system of equations (34) we have

$C_{z_{i}v}(t)=\langle z_{i}\,v\rangle_{\rm
eq}\,e^{-\lambda_{i}\,t}+e^{-\lambda_{i}\,t}*v(t)\,,\qquad i=1,\dots,n$ (36)

where we use the symbol “*” to indicate the convolution integral between any
two functions $f(t)*g(t)=\int_{0}^{t}f(t-\tau)\,g(\tau)\,d\tau$. Substituting
eqs. (36) into the first eq. (34) we obtain

$\frac{dC_{vv}(t)}{dt}=-h(t)*C_{vv}(t)-\sum_{i=1}^{n}e^{-\lambda_{i}\,t}\langle
z_{i}\,v\rangle_{\rm eq}$ (37)

In order to recover the fluctuation-dissipation relation of the first kind
(FD1k), that is a consequence of the Langevin condition, we should have
identically

$\langle z_{i}\,v\rangle_{\rm eq}=0\,,\qquad i=1,\dots,n$ (38)

In the initial-value representation of the FD theory, the system of equations
(38) represents the fundamental constraint imposed on the internal degrees of
freedom, out of which the equilibrium FD theory can be developed.
Specifically, from eq. (38), the expression for the intensity coefficients
$c_{i}$ directly follows after elementary algebraic manipulations of the
steady-state moment equations.

Appendix III - Stochastic realizations: the noncommutative case -  In the main
text we have considered the following kernel

$h(t)=a_{1}\,e^{-\lambda_{1}\,t}+a_{2}\,e^{-\lambda_{2}\,t}$ (39)

in nondimensional form (i.e. $m=1$ for the mass, and $\langle
v^{2}\rangle_{\rm eq}=1$ for the velocity), with $a_{1}=1$, $a_{2}=-\alpha$,
$\lambda_{1}=1/10$, $\lambda_{2}=1$. For positive values of $\alpha$,
$\bm{\Lambda}$ and $\bm{\beta}$ cannot commute to produce an equilibrium
behavior. Therefore, $\bm{\beta}$ should be a full matrix, and the local
representation follows

$\displaystyle\frac{dv(t)}{dt}$ $\displaystyle=$ $\displaystyle-
a_{1}\,z_{1}(t)-a_{2}\,z_{2}(t)$ $\displaystyle\frac{dz_{1}(t)}{dt}$
$\displaystyle=$
$\displaystyle-\lambda_{1}\,z_{1}(t)+v(t)+\sqrt{2}\,\left(\beta_{1,1}\,\xi_{1}(t)+\beta_{1,2}\,\xi_{2}(t)\right)$
(40) $\displaystyle\frac{dz_{2}(t)}{dt}$ $\displaystyle=$
$\displaystyle-\lambda_{2}\,z_{2}(t)+v(t)+\sqrt{2}\,\left(\beta_{2,1}\,\xi_{1}(t)+\beta_{2,2}\,\xi_{2}(t)\right)$

Observe that the stochastic forcing cannot act directly on the velocity
variable, as otherwise no equilibrium conditions could be achieved.

The Fokker-Planck equation for the density $p(v,z_{1},z_{2},t)$ associated
with eq. (40) is

$\frac{\partial p}{\partial t}=(a_{1}\,z_{1}+a_{2}\,z_{2})\frac{\partial
p}{\partial v}+\sum_{i=1}^{2}\lambda_{i}\frac{\partial(z_{i}\,p)}{\partial
z_{i}}-v\,\sum_{i=1}^{2}\frac{\partial p}{\partial
z_{i}}+\sum_{i=1}^{2}\sum_{j=1}^{2}S_{i,j}\,\frac{\partial^{2}p}{\partial
z_{i}\partial z_{j}}$ (41)

where $\bm{\beta}=(\beta_{i,j})_{i,j=1,2}$, and the matrix ${\bf
S}=(S_{i,j})_{i,j=1,2}$ is defined by

${\bf S}=\bm{\beta}\,\bm{\beta}^{T}$ (42)

where the superscript “$T$” indicate transpose. The matrix ${\bf S}$ should be
positive definite. Componentwise,

$S_{1,1}=\sum_{i=1}^{2}(\beta_{1,i})^{2}\,,\qquad
S_{2,2}=\sum_{i=1}^{2}(\beta_{2,i})^{2}\,,\quad
S_{1,2}=S_{2,1}=\sum_{i=1}^{2}\beta_{1,i}\,\beta_{2,i}$ (43)

In this case, the moment equations read

$\displaystyle\frac{d\langle v^{2}\rangle}{dt}$ $\displaystyle=$
$\displaystyle-a_{1}\,y_{1}-a_{2}y_{2}$ (44) $\displaystyle\frac{dy_{i}}{dt}$
$\displaystyle=$
$\displaystyle-\sum_{j=1}^{2}a_{j}m_{i,j}-\lambda_{i}\,y_{i}+\langle
v^{2}\rangle\,,\quad i=1,2$ (45) $\displaystyle\frac{dm_{i,j}}{dt}$
$\displaystyle=$
$\displaystyle-(\lambda_{i}+\lambda_{j})\,m_{i,j}+y_{i}+y_{j}+S_{i,j}+S_{j,i}\,,\quad
i,j=1,2$ (46)

where $y_{i}(t)=\langle z_{i}(t)\,v(t)\rangle$, $m_{i,j}(t)=\langle
z_{i}(t)\,z_{j}(t)\rangle$, $i,j=1,2$. Imposing the equilibrium and the
extended Langevin conditions eq. (38), i.e., $\langle v^{2}\rangle=\langle
v^{2}\rangle_{\rm eq}=1$, $y_{1}=y_{2}=0$, eq. (44) is identically satisfies,
while from eq. (46) we get

$m_{1,1}=\frac{S_{1,1}}{\lambda_{1}}\,,\quad
m_{1,2}=m_{2,1}=\frac{2\,S_{1,2}}{\lambda_{1}+\lambda_{2}}\,,\quad
m_{2,2}=\frac{S_{2,2}}{\lambda_{2}}$ (47)

Inserting the latter expressions for $m_{i,j}$ within eqs. (45), a linear
system in the ${\bf S}$-matrix entries is obtained

$\displaystyle
a_{1}\,\frac{S_{1,1}}{\lambda_{1}}+2\,a_{2}\,\frac{S_{1,2}}{\lambda_{1}+\lambda_{2}}$
$\displaystyle=$ $\displaystyle 1$ $\displaystyle
2\,a_{1}\,\frac{S_{1,2}}{\lambda_{1}+\lambda_{2}}+a_{2}\,\frac{S_{2,2}}{\lambda_{2}}$
$\displaystyle=$ $\displaystyle 1$ (48)

If $a_{2}>0$ (negative $\alpha$), we recover the commutative case, simply
setting $S_{1,2}=0$. For $a_{2}<0$, from eq. (48) we can explicit the diagonal
terms of the ${\bf S}$-matrix

$S_{1,1}=\frac{\lambda_{1}}{a_{1}}\,\left(1-\frac{2\,a_{2}}{\lambda_{1}+\lambda_{2}}\,S_{1,2}\right)\,,\qquad
S_{2,2}=-\frac{\lambda_{2}}{a_{2}}\left(\frac{2\,a_{1}}{\lambda_{1}+\lambda_{2}}\,S_{1,2}-1\right)$
(49)

Since $-\lambda_{2}/a_{2}>0$, the condition
$S_{1,2}>(\lambda_{1}+\lambda_{2})/2\,a_{1}$ implies that $S_{1,1}>0$,
$S_{2,2}>0$. In order to ensure the positive definite nature of ${\bf S}$ it
remains to fulfil the condition on its determinant, namely
$S_{1,1}\,S_{2,2}-\left(S_{1,2}\right)^{2}\geq 0$, that mathematically can be
regarded as a consequence of the Cauchy-Schwarz inequality. This leads to the
following condition expressed in terms of $S_{1,2}$

$\frac{\lambda_{1}}{a_{1}}\,\left(1-\frac{2\,a_{2}}{\lambda_{1}+\lambda_{2}}\,S_{1,2}\right)\,\left(-\frac{\lambda_{2}}{a_{2}}\right)\left(\frac{2\,a_{1}}{\lambda_{1}+\lambda_{2}}\,S_{1,2}-1\right)-S_{1,2}^{2}\geq
0$ (50)

Setting $\xi=S_{1,2}$, and thus $\xi>\xi_{\rm
min}=(\lambda_{1}+\lambda_{2})/2a_{1}$, eq. (50) can be compactly rewritten as
a quadratic inequality in $\xi$

$\phi(\xi)=-1+(\gamma_{2}-\gamma_{1})\,\xi-\left(\gamma_{3}-\gamma_{1}\,\gamma_{2}\right)\,\xi^{2}\geq
0$ (51)

where

$\gamma_{1}=-\frac{2\,a_{2}}{\lambda_{1}+\lambda_{2}}\,,\quad\gamma_{2}=\frac{2\,a_{1}}{\lambda_{1}+\lambda_{2}}\,,\quad\gamma_{3}=-\frac{a_{1}\,a_{2}}{\lambda_{1}\,\lambda_{2}}$
(52)

Since

$\gamma_{3}-\gamma_{1}\,\gamma_{2}=-\frac{a_{1}\,a_{2}\,(\lambda_{1}-\lambda_{2})^{2}}{\lambda_{1}\,\lambda_{2}(\lambda_{1}+\lambda_{2})^{2}}>0$
(53)

the coefficient of the quadratic term in $\xi$ of $\phi(\xi)$ is negative.
Consequently, in order to have stochastic realizability, the local maximum of
$\phi(\xi)$ at $\xi^{*}$ should be positive, $\phi(\xi^{*})>0$. From the
expression for $\phi(\xi)$ eq. (51), we have

$\xi^{*}=\frac{\gamma_{2}-\gamma_{1}}{2\,(\gamma_{3}-\gamma_{1}\,\gamma_{2})}\,,\qquad\phi(\xi^{*})=\frac{(\gamma_{1}+\gamma_{2})^{2}-4\,\gamma_{3}}{4\,(\gamma_{3}-\gamma_{1}\,\gamma_{2})}$
(54)

Therefore, if $(\gamma_{1}+\gamma_{2})^{2}-4\,\gamma_{3}>0$, the GLE is
stochastically realizable, and one can consider any value of $\xi$ falling in
the interval $(\xi_{-},\xi_{+})$, where

$\xi_{\pm}=\frac{(\gamma_{2}-\gamma_{1})}{2\,(\gamma_{3}-\gamma_{1}\,\gamma_{2})}\,\left[1\pm\sqrt{1-\frac{4(\gamma_{3}-\gamma_{1}\,\gamma_{2})}{(\gamma_{2}-\gamma_{1})}}\right]$
(55)

provided that these values are greater than $\xi_{\rm
min}=(\lambda_{1}+\lambda_{2})/2\,a_{1}$ (as is in the present case). Figure 4
depicts the behavior of $\phi(\xi^{*})$ for $a_{1}=1$, $\lambda_{1}=0.1$
$a_{2}=-\alpha$, $\lambda_{2}=1$, indicating that $\phi(\xi^{*})=0$ for
$\alpha=\alpha_{nc}=0.1$.

Figure 4:  $\phi(\xi^{*})$ vs $\alpha$ for $\alpha_{1}=1$, $\lambda_{1}=0.1$,
$\alpha_{2}=-\alpha$, $\lambda_{2}=1$.

Figure 5 depicts the behavior of $\xi_{+}$ and $\xi_{-}$ vs $\alpha$ in the
same conditions.

Figure 5: $\xi_{+}$ (upper curve) and $\xi_{-}$ (lower curve) vs $\alpha$, for
$\alpha_{1}=1$, $\lambda_{1}=0.1$, $\alpha_{2}=-\alpha$, $\lambda_{2}=1$. The
horizontal line corresponds to the value of $\xi_{\rm min}$.

Given $\alpha<\alpha_{\rm nc}$, any value of $\xi\in(\xi_{-},\xi_{+})$
produces a stochastic realization of the system at equilibrium, satisfying
either FD1k or FD2k in its strong form.

For $\alpha<\alpha_{nc}$, any value of $\xi$ in the interval
$(\xi_{-},\xi_{+})$ produces an equivalent model of the equilibrium behavior.
To this end, we have to show that all these realizations for
$\xi\in(\xi_{-},\xi_{+})$ give rise to one and the same autocorrelation
function of the resulting stochastic force at equilibrium (i.e. in the long-
term). To avoid misunderstandings, it is useful to remain that we are here
considering a nondimensional formulation with $m=1$, and $\langle
v^{2}\rangle_{\rm eq}=1$, and thus the FD2k result in the strong form
corresponds to $C_{RR}(t)=h(t)$. To verify this property consider the
expression for the stochastic force $R(t)$. From eq. (40) we have

$R(t)=\sum_{i=1}^{2}a_{i}\,z_{i,0}\,e^{\\-\lambda_{i}\,t}+\sqrt{2}\sum_{i=1}^{2}a_{i}\sum_{h=1}^{2}\,\beta_{i,h}\,e^{-\lambda_{i}t}*\xi_{h}(t)=2\sum_{i=1}^{2}a_{i}\sum_{h=1}^{2}\beta_{i,h}\,r_{i,h}(t)$
(56)

where we have set

$r_{i,h}(t)=e^{-\lambda_{i}t}*\xi_{h}(t)\,,\qquad i,h=1,2$ (57)

In the long-term (equilibrium), the first term at the r.h.s. of eq. (56)
decays exponentially to zero, so that

$C_{RR}(t)=\langle R(t)\,R(0)\rangle_{\rm
eq}=2\sum_{i=1}^{2}a_{i}\sum_{h=1}^{2}\beta_{i,h}\sum_{j=1}^{2}a_{j}\sum_{k=1}^{2}\beta_{j,k}\,\langle
r_{i,h}(t)\,r_{j,k}(0)\rangle_{\rm eq}$ (58)

where

$\langle r_{i,h}(t)\,r_{j,k}(0)\rangle_{\rm
eq}=\lim_{\tau\rightarrow\infty}\langle r_{i,h}(t+\tau)\,r_{j,k}(\tau)\rangle$
(59)

The latter expression can be easily calculated by quadraturae, enforcing the
property $\xi_{h}(t)\,\xi_{k}(\tau)=\delta_{h,k}\,\delta(t-\tau)$, and this
yields

$\langle r_{i,h}(t)\,r_{j,k}(0)\rangle_{\rm
eq}=\frac{e^{\lambda_{i}\,t}}{\lambda_{i}+\lambda_{j}}\,\delta_{h,k}$ (60)

Inserting the latter expression into eq. (58), we finally arrive to

$\displaystyle C_{RR}(t)$ $\displaystyle=$ $\displaystyle
2\sum_{i=1}^{2}a_{i}\sum_{h=1}^{2}\beta_{i,h}\sum_{j=1}^{2}a_{j}\sum_{k=1}^{2}\beta_{j,k}\,\frac{e^{\lambda_{i}\,t}}{\lambda_{i}+\lambda_{j}}\,\delta_{h,k}=2\sum_{i=1}^{2}a_{i}\sum_{j=1}^{2}\frac{a_{j}}{\lambda_{i}+\lambda_{j}}\,e^{-\lambda_{i}\,t}\sum_{h=1}^{2}\beta_{i,h}\,\beta_{j,h}$
(61) $\displaystyle=$
$\displaystyle\sum_{i=1}^{2}a_{i}\,e^{-\lambda_{i}\,t}\,\left[\sum_{j=1}^{1}2\,a_{j}\frac{S_{i,j}}{\lambda_{i}+\lambda_{i}}\right]$

The expression within square parentheses at the r.h.s. of eq. (61) is
identically equal to one for $i=1,2$ because of eqs. (48), and thus eq. (61)
simply becomes

$C_{RR}(t)=\sum_{i=1}^{2}a_{i}\,e^{-\lambda_{i}\,t}=h(t)$ (62)

which proves that for any plausible choice of $\xi$, all the stochastic
realizations are equivalent and satisfy FD2k.

Appendix IV - Linear response theory: exponential divergence outside
${\mathcal{D}}_{\rm ds}$ \-  Consider the case of “well-behaved kernels”
$h(t)$ possessing real and negative poles. In this case, the classical theory
predicts a bounded response in the presence of a periodically forced term
kubo2

$\frac{dv(t)}{dt}=-\int_{0}^{t}h(t-\tau)\,v(\tau)\,d\tau+F(t;\omega)$ (63)

where $F(t;\omega)$ is a sinusoidal perturbation of frequency $\omega$, say
$F(t;\omega)=F_{0}\,e^{\mathrm{i}\,\omega\,t}$, i.e.,
$v(t;\omega)=\mu(\omega)\,F_{0}\,e^{\mathrm{i}\,\omega\,t}$, controlled by the
mobility function $\mu(\omega)$ kubo2 ,

$\mu(\omega)=\frac{1}{\mathrm{i}\,\omega+h[\omega]}$ (64)

where $h[\omega]=\widehat{h}(\mathrm{i}\,\omega)$ is the Laplace-Fourier
transform of $h(t)$. This is not true if the GLE is not dissipatively stable,
for which the dynamics diverges exponentially in time in the presence of
generic sinusoidal perturbations, with an exponent controlled by the real part
of dominant eigenvalue, i.e. of the eigenvalue possessing the largest real
part.

Figure 6: $|v(t)|$ vs $t$ for sinusoidal forcings: (a):
$F(t,;\omega)=\cos(\omega\,t)$; (b): $F(t;\omega)=\sin(\omega\,t)$ at
$\omega=0.03$.

To show this, consider the kernel $h(t)$ given by eq. (39) at $\alpha=0.3$. In
this case, the eigenvalues of the matrix ${\bf A}$ associated with $h(t)$ are:
$\mu_{1,2}=\mu_{u}\pm\mathrm{i}\,\omega_{u}$, with $\mu_{u}\simeq
2.137\,\times 10^{-2}$, $\omega_{u}\simeq 0.921$, and
$\mu_{3}=\mu_{s}\simeq-1.14274$, corresponding to a couple of unstable complex
conjugate eigenvalues and a stable real one. The response to any perturbation
can be easily estimated in the Laplace domain (assuming $v(t=0)=0$),

$\displaystyle\widehat{v}(s)$ $\displaystyle=$
$\displaystyle\frac{1}{s+\widehat{h}(s)}\,\widehat{F}(s;\omega)$
$\displaystyle=$
$\displaystyle\frac{(s+\lambda_{1})\,(s+\lambda_{2})}{s\,(s+\lambda_{1})\,(s+\lambda_{2})+a_{1}\,(s+\lambda_{2})+a_{2}\,(s+\lambda_{1})}\,\widehat{F}(s;\omega)$
$\displaystyle=$
$\displaystyle\left[\frac{A+B\,s}{(s-\mu_{u})^{2}+\omega_{u}^{2}}+\frac{C}{s-\mu_{s}}\right]\,\widehat{F}(s;\omega)$

where $A,\,B\,,C$ can be easily calculated from the expansion in partial
fractions. For large $t$, solely the first term at the r.h.s. of eq. (Dynamic
fluctuation-dissipation theory for Generalized Langevin Equations:
constructive constraints, stability and realizability) is relevant and it
corresponds to a $v(t)$ exponentially growing in time with the exponent
$\mu_{\rm max}=\mu_{u}$ with a prefactor that depends on the frequency
$\omega$. Specifically

$v(t)=\mu_{c}^{(e)}(\omega)\,F_{0}\,e^{\mu_{u}\,t}\,\cos(\omega_{u}\,t+\varphi_{c})$
(66)

if $F(t;\omega)=F_{0}\,\cos(\omega\,t)$, and

$v(t)=\mu_{s}^{(e)}(\omega)\,F_{0}\,e^{\mu_{u}\,t}\,\cos(\omega_{u}\,t+\varphi_{s})$
(67)

if $F(t;\omega)=F_{0}\,\sin(\omega\,t)$, where $\mu_{c}^{(e)}(\omega)$ and
$\mu_{s}^{(e)}(\omega)$, referred to as the “exponential mobilities” can be
obtained analytically from eq. (Dynamic fluctuation-dissipation theory for
Generalized Langevin Equations: constructive constraints, stability and
realizability) in the two cases considered, and $\varphi_{c}$ and
$\varphi_{s}$ are the phase-shifts. Figure 6 shows some examples of this
exponential behavior, while the behavior of the exponential mobilities
$\mu_{c}^{(e)}(\omega)$ and $\mu_{s}^{(c)}(\omega)$ is reported in the main
text. For dissipatively unstable systems, apart from the exponential
divergence with time, the frequency of the oscillations is locked to
$\omega_{u}$, independently of $\omega$, marking the difference with respect
to the dissipatively stable case.

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††thanks: This work was partially supported by the European Union and the
European Social Fund through project Supercomputer, the national virtual lab
(grant no.: TÁMOP-4.2.2.C-11/1/KONV-2012-0010). This research was also
supported by the Hungarian Scientific Research Fund (OTKA) Grant NK 81402.

# Additive solvability and linear independence of the solutions of a system of
functional equations

Eszter Gselmann  Institute of Mathematics, University of Debrecen, H-4010
Debrecen, Pf. 12, Hungary<EMAIL_ADDRESS>Zsolt Páles  Institute
of Mathematics, University of Debrecen, H-4010 Debrecen, Pf. 12, Hungary
<EMAIL_ADDRESS>
###### Abstract.

The aim of this paper is twofold. On one hand, the additive solvability of the
system of functional equations

$\displaystyle
d_{k}(xy)=\sum_{i=0}^{k}\Gamma(i,k-i)d_{i}(x)d_{k-i}(y)\qquad(x,y\in\mathbb{R},\,k\in\\{0,\ldots,n\\})$

is studied, where $\Delta_{n}:=\big{\\{}(i,j)\in\mathbb{Z}\times\mathbb{Z}\mid
0\leq i,j\mbox{ and }i+j\leq n\big{\\}}$ and
$\Gamma\colon\Delta_{n}\to\mathbb{R}$ is a symmetric function such that
$\Gamma(i,j)=1$ whenever $i\cdot j=0$. On the other hand, the linear
dependence and independence of the additive solutions
$d_{0},d_{1},\dots,d_{n}\colon\mathbb{R}\to\mathbb{R}$ of the above system of
equations is characterized. As a consequence of the main result, for any
nonzero real derivation $d\colon\mathbb{R}\to\mathbb{R}$, the iterates
$d^{0},d^{1},\dots,d^{n}$ of $d$ are shown to be linearly independent, and the
graph of the mapping $x\mapsto(x,d^{1}(x),\dots,d^{n}(x))$ to be dense in
$\mathbb{R}^{n+1}$.

###### Key words and phrases:

derivation, higher order derivation, iterates, linear dependence

###### 1991 Mathematics Subject Classification:

Primary 16W25; Secondary 39B50

##  1\. Introduction

Given a real linear space $X$, a function $a\colon\mathbb{R}\to X$ is called
additive if

$\displaystyle a(x+y)=a(x)+a(y)\qquad(x,y\in\mathbb{R}).$ (1)

It is a nontrivial fact that additive functions may satisfy further functional
equations. Among these particular additive functions the so-called derivations
play an important role. An additive function $d\colon\mathbb{R}\to X$ is
called a derivation (cf. [6], [8]) if it satisfies the (first-order) Leibniz
Rule:

$\displaystyle d(xy)=xd(y)+yd(x)\qquad(x,y\in\mathbb{R}).$ (2)

Putting $x=y=1$ into (10), we get $d(1)=0$, hence, by the
$\mathbb{Q}$-homogeneity of additive functions, it follows that derivations
vanish at rational numbers. Therefore, assuming that $X$ is equipped with a
Hausdorff vector topology, the only continuous derivation is the identically
zero function. It can be shown that derivations with weak regularity
properties are necessarily continuous and consequently are identically equal
to zero. On the other hand, there exists derivations that are discontinuous
and henceforth very irregular (see [6]). More generally, for any algebraic
base $B$ of $\mathbb{R}$, and for any function $d_{0}\colon B\to X$, there
exists a unique derivation $d\colon\mathbb{R}\to X$ such that $d|_{B}=d_{0}$.

Given a real-valued derivation $d\colon\mathbb{R}\to\mathbb{R}$, one can prove
by induction that the iterates $d^{0}:=\mathop{\mbox{\rm id}}\nolimits$,
$d^{1}:=d$, …, $d^{n}:=d\circ d^{n-1}$ of $d$ satisfy the following higher-
order Leibniz Rule:

$\displaystyle
d^{k}(xy)=\sum_{i=0}^{k}\binom{k}{i}d^{i}(x)d^{k-i}(y)\qquad(x,y\in\mathbb{R},\,k\in\\{1,\ldots,n\\}).$
(3)

Motivated by this property, Heyneman–Sweedler [3] introduced the notion of
$n$th-order derivation (in the context of functions mapping rings to modules,
however, we will restrict ourselves only to real functions). Given
$n\in\mathbb{N}$, a sequence of additive functions
$d_{0},d_{1},\dots,d_{n}\colon\mathbb{R}\to\mathbb{R}$ is termed a derivation
of order $n$, if $d_{0}=\mathop{\mbox{\rm id}}\nolimits$ and, for any
$k\in\\{1,\ldots,n\\}$,

$\displaystyle
d_{k}(xy)=\sum_{i=0}^{k}\binom{k}{i}d_{i}(x)d_{k-i}(y)\qquad(x,y\in\mathbb{R})$
(4)

is fulfilled.

Clearly, a pair $(\mathop{\mbox{\rm id}}\nolimits,d)$ is a first-order
derivation if and only if $d$ is a derivation. More generally, if
$d\colon\mathbb{R}\to\mathbb{R}$ is a derivation, then the sequence
$(d^{0},d^{1},\dots,d^{n})$ is a derivation of order $n$. However, if
$\widetilde{d}\colon\mathbb{R}\to\mathbb{R}$ is a nontrivial derivation and
$n\geq 2$, then $(d^{0},d^{1},\dots,d^{n-1},d^{n}+\widetilde{d})$ is also an
$n$th-order derivation where the last element is not the $n$th iterate of the
derivation $d$.

The aim of this paper is twofold. On one hand, we study the additive
solvability of the following system of functional equations:

$\displaystyle
d_{k}(xy)=\sum_{i=0}^{k}\Gamma(i,k-i)d_{i}(x)d_{k-i}(y)\qquad(x,y\in\mathbb{R},\,k\in\\{0,\ldots,n\\}),$
(5)

where

$\displaystyle\Delta_{n}:=\big{\\{}(i,j)\in\mathbb{Z}\times\mathbb{Z}\mid
0\leq i,j\mbox{ and }i+j\leq n\big{\\}},$ (6)

and $\Gamma\colon\Delta_{n}\to\mathbb{R}$ is a symmetric function such that
$\Gamma(i,j)=1$ whenever $i\cdot j=0$. On the other hand, we characterize the
linear dependence and independence of the additive solutions
$d_{0},d_{1},\dots,d_{n}\colon\mathbb{R}\to\mathbb{R}$ of (5).

##  2\. On the additive solvability of the system of functional equations (5)

We recall first a particular case of the following result of Ebanks [1,
Theorem 3] (which generalizes a result of Jessen–Karpf–Thorup [4]):

######  Lemma 1.

Let $X$ be real linear space and $C,D\colon\mathbb{R}^{2}\to X$. Then there
exists a function $f\colon\mathbb{R}\to X$ such that

$\displaystyle C(x,y)$
$\displaystyle=f(x+y)-f(x)-f(y)\quad(x,y\in\mathbb{R}),$ (7) $\displaystyle
D(x,y)$ $\displaystyle=f(xy)-xf(y)-yf(x)\quad(x,y\in\mathbb{R})$

if and only if $C,D$ satisfy the following system of equations

$\displaystyle C(x+y,z)+C(x,y)$ $\displaystyle=C(x,y+z)+C(y,z)$
$\displaystyle(x,y,z\in\mathbb{R}),$ (8) $\displaystyle D(x,y)$
$\displaystyle=D(y,x)$ $\displaystyle(x,y\in\mathbb{R}),$ $\displaystyle
D(xy,z)+zD(x,y)$ $\displaystyle=D(x,yz)+xD(y,z)$
$\displaystyle(x,y,z\in\mathbb{R}),$ $\displaystyle C(xz,yz)-zC(x,y)$
$\displaystyle=D(x+y,z)-D(x,z)-D(y,z)$ $\displaystyle(x,y,z\in\mathbb{R}).$

As a trivial consequence of this result, we can characterize those two-
variable functions that are identical to the Leibniz difference of an additive
function.

######  Corollary 2.

Let $X$ be a real linear space and $D\colon\mathbb{R}^{2}\to X$. Then there
exists an additive function $f\colon\mathbb{R}\to X$ fulfilling functional
equation

$\displaystyle D(x,y)=f(xy)-xf(y)-yf(x)\qquad(x,y\in\mathbb{R})$ (9)

if and only if $D$ satisfies

$\displaystyle D(x,y)$ $\displaystyle=D(y,x)$
$\displaystyle\qquad(x,y\in\mathbb{R}),$ (10) $\displaystyle D(xy,z)+zD(x,y)$
$\displaystyle=D(x,yz)+xD(y,z)$ $\displaystyle\qquad(x,y,z\in\mathbb{R}),$
$\displaystyle D(x+y,z)$ $\displaystyle=D(x,z)+D(y,z)$
$\displaystyle\qquad(x,y,z\in\mathbb{R}).$

###### Proof.

Applying Lemma 1 for the function $C=0$, (7) is equivalent to the additivity
of $f$ and (9), and (8) reduces to (10). ∎

Our first main result offers a sufficient condition on the recursive additive
solvability of the functional equations (5). We deduce this result by using
Corollary 2, however, we note that another proof could be elaborated applyin
the results of Gselmann [2].

######  Theorem 3.

Let $n\geq 2$ and $\Gamma\colon\Delta_{n}\to\mathbb{R}$ be a symmetric
function such that $\Gamma(i,j)=1$ whenever $i\cdot j=0$ and

$\displaystyle\Gamma(i+j,k)\Gamma(i,j)=\Gamma(i,j+k)\Gamma(j,k)\quad(0\leq
i,j,k\text{ and }i+j+k\leq n).$ (11)

Let $d_{0}=\mathop{\mbox{\rm id}}\nolimits$ and let
$d_{1},\dots,d_{n-1}\colon\mathbb{R}\to\mathbb{R}$ be additive functions such
that (5) holds for $k\in\\{1,\dots,n-1\\}$. Then there exists an additive
function $d_{n}\colon\mathbb{R}\to\mathbb{R}$ such that (5) is also valid for
$k=n$.

###### Proof.

Using $\Gamma(0,n)=\Gamma(n,0)=1$, the functional equation for
$d_{n}\colon\mathbb{R}\to\mathbb{R}$ can be rewritten as

$\displaystyle d_{n}(xy)-xd_{n}(y)-yd_{n}(x)$ $\displaystyle=D_{n}(x,y)$ (12)
$\displaystyle:=\sum_{i=1}^{n-1}$
$\displaystyle\Gamma(i,n-i)d_{i}(x)d_{k-i}(y)\qquad(x,y\in\mathbb{R}).$

Thus, in view of Corollary 2, in order that there exist an additive function
$d_{n}$ such that (12) hold, it is necessary and sufficient that $D=D_{n}$
satisfy the conditions in (10). The symmetry of $\Gamma$ implies the symmetry,
the additivity of $d_{1},\dots,d_{n-1}$ results the biadditivity of $D_{n}$.
Thus, it suffices to prove that $D=D_{n}$ also satisfies the second identity
in (10). This is equivalent to showing that, for all fixed $y\in\mathbb{R}$,
the mapping $(x,z)\mapsto D_{n}(xy,z)+zD_{n}(x,y)$ is symmetric. Using
equations (5) for $k\in\\{1,\dots,n-1\\}$, we obtain

$\displaystyle D_{n}(xy,z)$ $\displaystyle+zD_{n}(x,y)$
$\displaystyle=\sum_{k=1}^{n-1}\Gamma(k,n-k)d_{k}(xy)d_{n-k}(z)+z\sum_{i=1}^{n-1}\Gamma(i,n-i)d_{i}(x)d_{n-i}(y)$
$\displaystyle=\sum_{k=1}^{n-1}\Gamma(k,n-k)\bigg{(}\sum_{i=0}^{k}\Gamma(i,k-i)d_{i}(x)d_{k-i}(y)\bigg{)}d_{n-k}(z)$
$\displaystyle\hskip
56.9055pt+z\sum_{i=1}^{n-1}\Gamma(i,n-i)d_{i}(x)d_{n-i}(y)$
$\displaystyle=\sum_{k=0}^{n}\sum_{i=0}^{k}\Gamma(k,n-k)\Gamma(i,k-i)d_{i}(x)d_{k-i}(y)d_{n-k}(z)$
$\displaystyle\hskip 56.9055pt-xyd_{n}(z)-xzd_{n}(y)-yzd_{n}(x)$
$\displaystyle=\sum_{\alpha,\beta,\gamma\geq
0,\,\alpha+\beta+\gamma=n}\Gamma(\alpha+\beta,\gamma)\Gamma(\alpha,\beta)d_{\alpha}(x)d_{\beta}(y)d_{\gamma}(z)$
$\displaystyle\hskip 56.9055pt-xyd_{n}(z)-xzd_{n}(y)-yzd_{n}(x).$

The sum of the last three terms in the above expression is symmetric in
$(x,z)$. The symmetry of the first summand is the consequence of the symmetry
of $(\alpha,\gamma)\mapsto\Gamma(\alpha+\beta,\gamma)\Gamma(\alpha,\beta)$
which follows from property (11). ∎

In what follows, we describe the nowhere zero solutions of (11).

######  Theorem 4.

Let $n\geq 2$ and $\Gamma\colon\Delta_{n}\to\mathbb{R}\setminus\\{0\\}$ be a
symmetric function so that $\Gamma(i,j)=1$ whenever $i\cdot j=0$. Then
$\Gamma$ satisfies the functional equation (11) if and only if there exists a
function $\gamma\colon\\{0,1,\ldots,n\\}\to\mathbb{R}\setminus\\{0\\}$ such
that

$\displaystyle\Gamma(i,j)=\frac{\gamma(i+j)}{\gamma(i)\gamma(j)}\qquad((i,j)\in\Delta_{n}).$
(13)

###### Proof.

Define the function
$\gamma\colon\\{0,1,\ldots,n\\}\to\mathbb{R}\setminus\\{0\\}$ through

$\displaystyle\gamma(k)=\prod_{\ell=1}^{k-1}\Gamma(\ell,1)\qquad(k\in\\{0,1,\ldots,n\\}).$

The empty product being equal to $1$, we have that $\gamma(0)=\gamma(1)=1$.

To complete the proof, we have to show that, for any $(i,j)\in\Delta_{n}$,

$\displaystyle\Gamma(i,j)=\frac{\gamma(i+j)}{\gamma(i)\gamma(j)}.$

This equivalent to proving that

$\displaystyle\Gamma(i,j)\prod_{\ell=1}^{i-1}\Gamma(\ell,1)=\prod_{\ell=j}^{i+j-1}\Gamma(\ell,1)\qquad((i,j)\in\Delta_{n}).$
(14)

This identity trivially holds for $i=0$, $i=1$ and for any
$j\in\\{0,\dots,n-i\\}$. Let $j\in\\{0,\dots,n-2\\}$ be fixed. We prove (14)
by induction on $i\in\\{1,\dots,n-j\\}$. Assume that (14) holds for
$i\in\\{1,\dots,n-j-1\\}$. Then,

$\displaystyle\Gamma(i+1,j)\prod_{\ell=1}^{i}\Gamma(\ell,1)$
$\displaystyle=\frac{\Gamma(i+1,j)\Gamma(i,1)}{\Gamma(i,j)}\bigg{(}\Gamma(i,j)\prod_{\ell=1}^{i-1}\Gamma(\ell,1)\bigg{)}$
(15)
$\displaystyle=\frac{\Gamma(i+1,j)\Gamma(i,1)}{\Gamma(i,j)}\prod_{\ell=j}^{i+j-1}\Gamma(\ell,1)$
$\displaystyle=\frac{\Gamma(i+1,j)\Gamma(i,1)}{\Gamma(i,j)\Gamma(i+j,1)}\prod_{\ell=j}^{i+j}\Gamma(\ell,1).$

Using (11), it follows that
$\Gamma(i+1,j)\Gamma(i,1)=\Gamma(i,j)\Gamma(i+j,1)$, hence (15) yields (14)
for $i+1$ instead of $i$.

Conversely, suppose that there exists a function
$\gamma\colon\\{0,1,\ldots,n\\}\to\mathbb{R}\setminus\\{0\\}$ such that

$\displaystyle\Gamma(i,j)=\frac{\gamma(i+j)}{\gamma(i)\gamma(j)}\qquad((i,j)\in\Delta_{n}).$

Then, for any $i,j,k\geq 0$ with $i+j+k\leq n$, we have

$\displaystyle\Gamma(i+j,k)\Gamma(i,j)$
$\displaystyle=\frac{\gamma(i+j+k)}{\gamma(i+j)\gamma(k)}\cdot\frac{\gamma(i+j)}{\gamma(i)\gamma(j)}$
$\displaystyle=\frac{\gamma(i+j+k)}{\gamma(i)\gamma(j+k)}\cdot\frac{\gamma(j+k)}{\gamma(j)\gamma(k)}=\Gamma(i,j+k)\Gamma(j,k),$

which completes the proof. ∎

When $\Gamma$ is of the form (13), then Theorem 3 reduces to the following
statement.

######  Corollary 5.

Let $n\geq 2$ and
$\gamma\colon\\{0,1,\ldots,n\\}\to\mathbb{R}\setminus\\{0\\}$ with
$\gamma(0)=1$. Let $d_{0}=\mathop{\mbox{\rm id}}\nolimits$ and let
$d_{1},\dots,d_{n-1}\colon\mathbb{R}\to\mathbb{R}$ be additive functions such
that

$\displaystyle
d_{k}(xy)=\sum_{i=0}^{k}\frac{\gamma(k)}{\gamma(i)\gamma(k-i)}d_{i}(x)d_{k-i}(y)\qquad(x,y\in\mathbb{R})$
(16)

holds for $k\in\\{1,\dots,n-1\\}$. Then there exists an additive function
$d_{n}\colon\mathbb{R}\to\mathbb{R}$ such that (16) is also valid for $k=n$.

We note that if in the above corollary $\gamma(k)=k!$, then (16) is equivalent
to (4), that is $\mathop{\mbox{\rm id}}\nolimits,d_{1},\dots,d_{n}$ is a
derivation of order $n$.

##  3\. A characterization of the linear dependence of additive functions

######  Theorem 6.

Let $X$ be a Hausdorff locally convex linear space and let
$a\colon\mathbb{R}\to X$ be an additive function. Then the following
statements are equivalent:

  1. (i)

There exists a nonzero continuous linear functional $\varphi\in X^{*}$ such
that $\varphi\circ a=0$;

  2. (ii)

There exists an upper semicontinuous function $\Phi:X\to\mathbb{R}$ such that
$\Phi\not\geq 0$ and $\Phi\circ a\geq 0$;

  3. (iii)

The range of $a$ is not dense in $X$, i.e., $\overline{a(\mathbb{R})}\neq X$.

###### Proof.

The implication (i)$\Rightarrow$(ii) is obvious, because $\Phi$ can be chosen
as $\varphi$.

To prove (ii)$\Rightarrow$(iii), assume that there exists an upper
semicontinuous function $\Phi:X\to\mathbb{R}$ such that $\Phi\not\geq 0$ and
$\Phi\circ a\geq 0$. Then $U:=\\{x\in X\mid\Phi(x)<0\\}$ is a nonempty and
open set. The inequality $\Phi\circ a\geq 0$ implies that $U\cap
a(\mathbb{R})=\emptyset$, which proves that the range of $a$ cannot be dense
in $X$.

Finally, suppose that $\overline{a(\mathbb{R})}\neq X$. By the additivity of
$a$, the set $a(\mathbb{R})$ is closed under addition and multiplication by
rational numbers. Therefore, the closure of $a(\mathbb{R})$ is a proper closed
linear subspace of $X$. Then, by the Hahn–Banach theorem, there exists a
nonzero continuous linear functional $\varphi\in X^{*}$ which vanishes on
$a(\mathbb{R})$, i.e., $\varphi\circ a=0$ is satisfied. ∎

By taking $X=\mathbb{R}^{n}$, the above theorem immediately simplifies to the
following consequence which characterizes the linear dependence of finitely
many additive functions.

######  Corollary 7.

Let $n\in\mathbb{N}$ and $a_{1},\dots,a_{n}\colon\mathbb{R}\to\mathbb{R}$ be
additive functions. Then the following statements are equivalent:

  1. (i)

The additive functions $a_{1},\dots,a_{n}$ are linearly dependent, i.e., there
exist $c_{1},\dots,c_{n}\in\mathbb{R}$ such that
$c_{1}^{2}+\cdots+c_{n}^{2}>0$ and $c_{1}a_{1}+\dots+c_{n}a_{n}=0$;

  2. (ii)

There exists an upper semicontinuous function
$\Phi\colon\mathbb{R}^{n}\to\mathbb{R}$ such that $\Phi\not\geq 0$ and

$\displaystyle\Phi(a_{1}(x),\dots,a_{n}(x))\geq 0\qquad(x\in\mathbb{R});$

  3. (iii)

The set $\\{(a_{1}(x),\dots,a_{n}(x))\mid x\in\mathbb{R}\\}$ is not dense in
$\mathbb{R}^{n}$.

In the particular case of this corollary, namely when $\Phi$ is an indefinite
quadratic form, the equivalence of statements (i) and (ii) is the main result
of the paper [5] by Kocsis. A former result in this direction is due to Maksa
and Rätz [7]: If two additive functions $a,b\colon\mathbb{R}\to\mathbb{R}$
satisfy $a(x)b(x)\geq 0$ then $a$ and $b$ are linearly dependent.

##  4\. Linear independence of iterates of nonzero derivations

In this section we apply Corollary 7 to the particular case when the additive
functions are iterates of a real derivation. However, firstly we prove the
following for higher order derivations.

######  Theorem 8.

Let $n\in\mathbb{N}$, let $\Gamma\colon\Delta_{n}\to\mathbb{R}$ be a symmetric
function such that $\Gamma(i,j)=1$ whenever $i\cdot j=0$, (11) is satisfied
and, for all $k\in\\{2,\dots,n\\}$ there exists $i\in\\{1,\dots,k-1\\}$ such
that $\Gamma(i,k-i)\neq 0$. Assume that $d_{0}=\mathop{\mbox{\rm
id}}\nolimits$ and $d_{1},\dots,d_{n}\colon\mathbb{R}\to X$ are additive
functions satisfying (5) for all $k\in\\{1,\dots,n\\}$. Then the following
statements are equivalent:

  1. (i)

There exist $c_{0},c_{1},\dots,c_{n}\in\mathbb{R}$ such that
$c_{0}^{2}+c_{1}^{2}+\cdots+c_{n}^{2}>0$ and

$\displaystyle
c_{0}x+c_{1}d_{1}(x)+\cdots+c_{n}d_{n}(x)=0\qquad(x\in\mathbb{R});$ (17)

  2. (ii)

There exists an upper semicontinuous function
$\Phi\colon\mathbb{R}^{n+1}\to\mathbb{R}$ such that $\Phi\not\geq 0$ and

$\displaystyle\Phi(x,d_{1}(x),\dots,d_{n}(x))\geq 0\qquad(x\in\mathbb{R});$

  3. (iii)

The set $\\{(x,d_{1}(x),\dots,d_{n}(x))\mid x\in\mathbb{R}\\}$ is not dense in
$\mathbb{R}^{n+1}$;

  4. (iv)

$d_{1}=0$.

###### Proof.

Applying Corollary 7 to the additive functions $a_{i}(x)=d_{i}(x)$
$(i\in\\{0,1,\dots,n\\})$, it follows that (i), (ii) and (iii) are equivalent.
The implication (iv)$\Rightarrow$(i) is obvious since if $d_{1}=0$, then (i)
holds with $c_{1}=1$ and $c_{0}=c_{2}=\cdots=c_{n}=0$.

Thus, it remains to show that (i) implies (iv). Assume that (i) holds. Then
there exist a smallest $1\leq m\leq n$ and $c_{0},\dots,c_{m}\in\mathbb{R}$
such that $c_{0}^{2}+c_{1}^{2}+\cdots+c_{m}^{2}>0$ and

$\displaystyle
c_{0}x+c_{1}d_{1}(x)+\cdots+c_{m}d_{m}(x)=0\qquad(x\in\mathbb{R}).$ (18)

This means that the equality

$\displaystyle\gamma_{0}x+\gamma_{1}d_{1}(x)+\cdots+\gamma_{m-1}d_{m-1}(x)=0\qquad(x\in\mathbb{R})$

can only hold for $\gamma_{0}=\cdots=\gamma_{m-1}=0$.

Observe, that $d_{1}(1)=\cdots=d_{n}(1)=0$. Indeed, $d_{1}(1)=0$ is a
consequence of (5) when $k=1$ because this equation means that $d_{1}$ is a
derivation. The rest easily follows by induction on $k$ from (5).

Putting $x=1$ into (18), it follows that $c_{0}=0$. If $m=1$, then $c_{1}$
cannot be zero, hence we obtain that $d_{1}=0$. Thus, we may assume that the
minimal $m$ for which (18) is satisfied is non-smaller than $2$. Replacing $x$
by $xy$ in (18) and applying (5), for all $x,y\in\mathbb{R}$, we get

$\displaystyle 0$
$\displaystyle=\sum_{k=1}^{m}c_{k}d_{k}(xy)=\sum_{k=1}^{m}c_{k}\bigg{(}\sum_{i=0}^{k}\Gamma(i,k-i)d_{i}(x)d_{k-i}(y)\bigg{)}$
$\displaystyle=\sum_{k=2}^{m}c_{k}\bigg{(}\sum_{i=1}^{k-1}\Gamma(i,k-i)d_{i}(x)d_{k\\!-\\!i}(y)\bigg{)}\\!+\\!x\bigg{(}\sum_{k=1}^{m}c_{k}d_{k}(y)\bigg{)}\\!+\\!y\bigg{(}\sum_{k=1}^{m}c_{k}d_{k}(x)\bigg{)}$
$\displaystyle=\sum_{k=2}^{m}\sum_{i=1}^{k-1}c_{k}\Gamma(i,k-i)d_{i}(x)d_{k-i}(y)=\sum_{i=1}^{m-1}\sum_{k=i+1}^{m}c_{k}\Gamma(i,k-i)d_{i}(x)d_{k-i}(y)$
$\displaystyle=\sum_{i=1}^{m-1}\bigg{(}\sum_{j=1}^{m-i}c_{i+j}\Gamma(i,j)d_{j}(y)\bigg{)}d_{i}(x).$

By the minimality of $m$, it follows from the above equality that, for all
$y\in\mathbb{R}$,

$\displaystyle\sum_{j=1}^{m-i}c_{i+j}\Gamma(i,j)d_{j}(y)=0\qquad(i\in\\{1,\dots,m-1\\}).$

Again, by the minimality of $m$, this implies that $c_{i+j}\Gamma(i,j)=0$ for
$(i,j)\in\Delta_{m}$ with $i,j\geq 1$. By the assumption of the theorem, for
all $k\in\\{2,\dots,n\\}$ there exists $i\in\\{1,\dots,k-1\\}$ such that
$\Gamma(i,k-i)\neq 0$. Thus, $c_{2}=\dots=c_{m}=0$. Therefore, by (18),
$c_{1}$ cannot be equal to zero. Then (18) simplifies to $d_{1}=0$, which was
to be proved. ∎

Let $n\in\mathbb{N}$ be arbitrary and $d\colon\mathbb{R}\to\mathbb{R}$ be a
derivation. Then the $(n+1)$-tuple $(\mathop{\mbox{\rm
id}}\nolimits,d,d^{2},\dots,d^{n})$ is a derivation of order $n$. Thus from
the previous theorem we immediately get the following.

######  Corollary 9.

Let $n\in\mathbb{N}$ and let $d\colon\mathbb{R}\to\mathbb{R}$ be a derivation.
Then the following statements are equivalent:

  1. (i)

There exist $c_{0},c_{1},\dots,c_{n}\in\mathbb{R}$ such that
$c_{0}^{2}+c_{1}^{2}+\cdots+c_{n}^{2}>0$ and

$\displaystyle c_{0}x+c_{1}d(x)+\cdots+c_{n}d^{n}(x)=0\qquad(x\in\mathbb{R});$
(19)

  2. (ii)

There exists an upper semicontinuous function
$\Phi\colon\mathbb{R}^{n+1}\to\mathbb{R}$ such that $\Phi\not\geq 0$ and

$\displaystyle\Phi(x,d(x),\dots,d^{n}(x))\geq 0\qquad(x\in\mathbb{R});$

  3. (iii)

The set $\\{(x,d(x),\dots,d^{n}(x))\mid x\in\mathbb{R}\\}$ is not dense in
$\mathbb{R}^{n+1}$;

  4. (iv)

$d=0$.

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