Characterising the nonequilibrium stationary states of Ornstein-Uhlenbeck processes
We characterise the nonequilibrium stationary state of a generic multivariate Ornstein-Uhlenbeck process involving degrees of freedom. The irreversibility of the process is encoded in the antisymmetric part of the Onsager matrix. The linearity of the Langevin equations allows us to derive closed-form expressions in terms of the latter matrix for many quantities of interest, in particular the entropy production rate and the fluctuation-dissipation ratio matrix. This general setting is then illustrated by two classes of systems. First, we consider the one-dimensional ferromagnetic Gaussian spin model endowed with a stochastic dynamics where spatial asymmetry results in irreversibility. The stationary state on a ring is independent of the asymmetry parameter, whereas it depends continuously on the latter on an open chain. Much attention is also paid to finite-size effects, especially near the critical point. Second, we consider arrays of resistively coupled electrical circuits. The entropy production rate is evaluated in the regime where the local temperatures of the resistors have small fluctuations. For networks the entropy production rate grows linearly with the size of the array. For networks a quadratic growth law violating extensivity is predicted.
The everlasting interest for nonequilibrium phenomena in Statistical Physics has experienced a considerable upsurge in the last decades with the emergence of a series of general results known as fluctuation theorems. These theoretical developments have emphasised the central role of the rate of entropy production per unit time, which is the key quantity involved in the Gallavotti-Cohen fluctuation theorem  (see also [2, 3, 4, 5, 6, 7]). Even though the entropy production rate appears as a fundamental quantity characterising nonequilibrium stationary states, the latter possess many other features of interest, including a violation of the fluctuation-dissipation theorem characteristic of equilibrium states in the linear-response regime and non-local fluctuations in the large-deviation regime [8, 9, 10, 11].
The goal of this work is to investigate the most salient characteristic features of nonequilibrium stationary states within a unified framework in a specific class of models which lend themselves to an analytic approach, namely linear Langevin systems. These systems can be viewed as higher-dimensional extensions of the well-known Ornstein-Uhlenbeck process , describing the velocity of a Brownian particle in one dimension,
where is the friction coefficient and is a Gaussian white noise with amplitude , i.e.,
The system equilibrates under the combined effects of linear damping and noise. The equilibrium probability distribution of the velocity is the Maxwell-Boltzmann distribution, i.e., a Gaussian law with variance . The Ornstein-Uhlenbeck process is reversible and obeys detailed balance with respect to the above distribution. The Einstein relation is an example of an equilibrium fluctuation-dissipation relation.
Generalising the Ornstein-Uhlenbeck process to several linearly coupled degrees of freedom appears as a very natural construction. The first occurrence of a multivariate Ornstein-Uhlenbeck process can be traced back to . Such a process is defined by linear Langevin equations describing the evolution of coupled dynamical variables. Remarkably enough, in stark contrast with the historical case recalled above, a generic multivariate Ornstein-Uhlenbeck process is irreversible [14, 15, 16, 17, 18]. To close this introduction of the problem, let us mention related works concerning the structure of either dissipative or otherwise irreversible dynamical systems near fixed points, where the dynamical equations can be linearised [19, 20, 21, 22]. In the stochastic setting, this line of thought leads to consider multivariate Ornstein-Uhlenbeck processes.
This paper is devoted to a systematic characterisation of the nonequilibrium stationary states of multivariate Ornstein-Uhlenbeck processes. The linearity of the Langevin equations governing these processes allows us to use tools from linear algebra. In section 2, devoted to the general framework, we demonstrate that the amount of irreversibility is measured by the antisymmetric part of the Onsager matrix . Multivariate Ornstein-Uhlenbeck processes are defined in section 2.1, the Sylvester equation governing their steady state is derived in section 2.2, and their irreversibility is characterised in section 2.3. The various quantities characterising their nonequilibrium stationary state are then derived in terms of the matrix or of its associated Hermitian form : stationary probability current (section 2.4), entropy production rate (section 2.5), correlation, response and fluctuation-dissipation ratio (section 2.6). Various other facets of the problem are also investigated, in particular the effect of a linear change of coordinates (section 2.7), a formal investigation of the case where the friction matrix (see (2.2)) is diagonalisable (section 2.8), cyclically symmetric processes (section 2.9) and the case of two degrees of freedom (section 2.10).
We then analyse in detail two different examples of physical systems giving rise to multivariate Ornstein-Uhlenbeck processes. Our first example (sections 3 and 4) is borrowed from the microscopic world of kinetic spin systems. We consider the one-dimensional ferromagnetic Gaussian spin model endowed with a stochastic dynamics where spatial asymmetry results in irreversibility, along the lines of earlier studies on the Ising and spherical ferromagnets [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34]. The geometry of a finite ring is discussed in section 3. In this situation, the asymmetric dynamics corresponds to a cyclically symmetric process, and the stationary state is independent of the asymmetry parameter . We successively investigate the statics of the model (section 3.1), the conventional gradient dynamics (section 3.2), the asymmetric dynamics (section 3.3), and the key observables characterising the resulting nonequilibrium stationary state, first in the thermodynamic limit (section 3.4) and then on finite-size systems (section 3.5). Section 4 is devoted to the model on a finite open chain. There, at variance with the ring geometry, the stationary state depends continuously on the asymmetry parameter . We study the statics of the model (section 4.1), general properties of the asymmetric dynamics (section 4.2), the spectrum of the friction matrix (section 4.3), the first few system sizes (section 4.4), and the main features of the nonequilibrium stationary state (section 4.5). We then turn to more detailed investigations of the model, both in the small- regime (section 4.6) and in the totally asymmetric case (section 4.7). Our second class of examples of multivariate Ornstein-Uhlenbeck processes originates in macroscopic physics. We consider arrays of resistively coupled electrical circuits, where resistors are sources of thermal noise. Within this setting, irreversibility is brought about by an inhomogeneous temperature profile. The main emphasis is on the entropy production rate per unit time in the regime where the fluctuations of the temperature profile are small. We successively consider networks (section 5.1) and networks (section 5.2). The scaling of the entropy production rate with the network size is very different in both cases. Section 6 contains a brief discussion of our findings. Two appendices contain the derivations of the entropy production amplitude on the open spin chain (A) and of the amplitude matrices giving the entropy production rates on both types of electrical arrays (B).
2 General framework
2.1 The model
(), i.e., in vector and matrix notations,
The are Gaussian white noises such that
Throughout this work boldface symbols denote vectors and matrices, and the superscript denotes the transpose of a vector or of a matrix.
The process under study is thus defined by two real matrices of size : the noise covariance matrix or diffusion matrix , which is symmetric by construction, and the friction matrix , which is not symmetric in general. In what follows, we assume that is a positive definite matrix, and that is the opposite of a stability (or Hurwitz) matrix. This means that all its eigenvalues , which are in general complex, obey
We introduce for further reference the spectral gap
The above assumptions exclude degenerate cases. They ensure that the process relaxes exponentially fast to in the absence of noise, and to a fluctuating stationary state with Gaussian statistics in the presence of noise. The main goal of this work is to characterise various features of the latter state, which is generically a nonequilibrium stationary state.
2.2 Relaxation and stationary state
The relaxation of the process from its (deterministic) initial value is encoded in the Green’s function
with . The hypothesis made on the friction matrix ensures that falls off exponentially fast to zero. We have
The process is Gaussian, as it can be expressed linearly in terms of the Gaussian white noises . Its statistics at any given time is entirely characterised by its mean value, , and by its full equal-time correlation matrix
Equation (2.9) implies
It can be checked that obeys the differential equation
The stationary state of the process is therefore Gaussian, with zero mean, and characterised by the stationary covariance matrix
This stationary state is therefore characterised by the probability density
Equation (2.12) implies
This is the key equation of the problem, relating the stationary covariance matrix to the friction and diffusion matrices and defining the process. Linear matrix equations of this type are referred to as Sylvester (or Lyapunov) equations.
2.3 Reversible vs. irreversible processes
In this situation, each term in the left-hand side of (2.15) is separately equal to . The stationary covariance matrix reads
The corresponding stationary state is an equilibrium state.
Whenever the symmetry condition (2.16) is not obeyed, the process is irreversible, with a nonequilibrium stationary state. Solving the Sylvester equation (2.15) for the covariance matrix is more difficult than inverting the friction matrix . We parametrise the matrices , and by setting
The matrix is the Onsager matrix of kinetic coefficients [2, 18]. Its antisymmetric part provides a measure of the amount of irreversibility of the process. If the process is reversible, is symmetric and vanishes.
It is advantageous to recast the matrix in the following balanced form:
where is the imaginary unit, whereas , the positive square root of the diffusion matrix , is a symmetric positive definite matrix. The matrix thus defined is both dimensionless and Hermitian. Its entries are purely imaginary, and so is a real antisymmetric matrix. As a consequence, the eigenvalues of occur in pairs of opposite real numbers (unless ). Its spectral radius, i.e., its largest positive eigenvalue,
2.4 Stationary probability current
where the stationary probability density is given by (2.14). We thus obtain
where the mobility tensor reads (see (2.18))
The stationary probability current is therefore proportional to , the matrix measuring the irreversibility of the process.
2.5 Entropy production rate
Associating an increase of entropy to an irreversible process is one of the possible statements of the second law of Thermodynamics. In more recent times, much attention has been devoted to the rate of entropy production per unit time for an open system [38, 39]. The latter rate is also the key quantity involved in the Gallavotti-Cohen fluctuation theorem . By now, it is commonly recognised as being a fundamental quantity characterising a nonequilibrium stationary state.
In the present context of diffusion processes, a general expression for the entropy production rate per unit time in the stationary state seemingly appears for the first time in print in  (see also [22, 41, 42, 43, 44] and [45, 46]). In our notations, this reads
As expected, is strictly positive if the process is irreversible, and it vanishes only if the process is reversible. Furthermore, since the stationary state is Gaussian with covariance matrix , we have , and so
These formulas express the entropy production rate as a quadratic form in the matrices , and characterising the irreversibility of the process. Finally, using again (2.18), (2.28) can be recast into the following expressions
involving neither the covariance matrix nor its inverse explicitly.
2.6 Correlation, response and fluctuation-dissipation ratio
We now turn to an investigation of the dynamics of the process, both in its transient regime and in its stationary state. Keeping in line with recent studies of slow dynamics and aging phenomena (see [48, 49, 50] for reviews), we focus our attention onto the correlation, response and fluctuation-dissipation ratio.
The correlation matrix is defined as
where the two times and are such that . This quantity obeys
with initial value for , hence (see (2.7))
The response matrix is defined as
where the ordering fields are linearly coupled to the dynamical variables , i.e., they are added to the noises on the right-hand side of (2.1). We thus readily obtain
In the stationary state, keeping the time lag fixed, the stationary correlation and response matrices read
We have therefore
The latter formula can be reshaped by introducing a dimensionless stationary fluctuation-dissipation ratio (FDR) matrix , such that
This definition agrees with the usage in the literature on slow dynamics and aging phenomena [50, 51]. The diffusion matrix plays the role of temperature. The fact that and occur in the rightmost positions on both sides of (2.37) is due to the conventions used in the definitions (2.30), (2.33) of the correlation and response matrices. We thus obtain
In the case of a reversible process (), we have and the equilibrium fluctuation-dissipation theorem is recovered as
In the general situation of an irreversible process, the FDR matrix is non-trivial. The second expression of (2.39) shows that its eigenvalues are of the form
where are the eigenvalues of , introduced below (2.19). The therefore lie on the circle with diameter [0,1] in the complex plane, and they occur in complex conjugate pairs (unless ). The center of mass of these eigenvalues defines the typical FDR
This quantity is real and such that , the upper bound being attained for reversible processes. Figure 1 shows a sketch of the spectrum of the FDR matrix in a typical situation with .
2.7 Considerations on reparametrisation
In this section we investigate the effect of a reparametrisation, i.e., of a change of coordinates. In order to keep the linearity of the process (2.1), we restrict ourselves to a linear change of coordinates of the form
where is an invertible matrix.
In terms of the new coordinates , the process is transformed into a similar one, characterised by the new matrices
The matrices and transform as , i.e.,
whereas the matrices and transform as
The matrices , and are respectively conjugate to , and , so that their spectra are invariant under reparametrisation. The entropy production rate , the typical FDR and the asymmetry index are also invariant under reparametrisation. This corroborates the intrinsic nature of these quantities.
It is worth investigating to what extent the process (2.1) can be simplified by means of a suitably chosen linear reparametrisation. Not much can be gained on the side of the friction matrix, as and have the same spectrum. This invariance could be expected as well, as the spectrum of encodes the relaxation times of the process. The diffusion matrix can however, at least in principle, always be brought to the canonical form
corresponding to independent normalised white noises, by choosing .
2.8 The case where the friction matrix is diagonalisable
As already stated, solving the Sylvester equation (2.15) for the covariance matrix is more difficult than inverting the friction matrix . The generic situation where is diagonalisable can however be formally dealt with as follows . Assume that has a biorthogonal system of left and right eigenvectors such that and , in Dirac notation, with
We have then
Equation (2.11) implies
In particular, the stationary covariance matrix reads
After reduction the denominator of this expression reads
The first factor, equal to , has degree in the entries of , whereas the second factor has degree . Altogether, is a polynomial of degree . This degree equals the number of independent entries in the symmetric matrix .
With the same conventions, we have
When the process is reversible, i.e., if (2.16) holds, it can be checked that the matrix element vanishes whenever and are different, and so vanishes, as should be.
This scheme will be used several times in the course of this work, starting with section 2.9. In the happy instances where either and commute or is simple enough, quantities of physical interest such as can also be expressed in terms of the eigenvectors of .
2.9 Cyclic symmetry
As usual, the complexity of the problem can be reduced in the presence of symmetries. In this section we consider cyclically symmetric processes, where the dynamical variables live on the sites labelled of a ring of points and the dynamics is invariant under discrete translations along the ring. The matrices and are therefore circulant matrices, whose entries and only depend on the difference mod . For instance, for this reads
The property that is symmetric imposes .
In this setting, it is natural to use the discrete Fourier transform
We recall that the indices and are understood mod . If needed, they can therefore be restricted to the range . Cyclic symmetry brings out a considerable simplification. All matrices pertaining to the problem are diagonal in the Fourier basis, and therefore commute with each other. The eigenvalues of a cyclically symmetric matrix indeed coincide with its discrete Fourier amplitudes .
The diffusion matrix is symmetric, i.e., we have , and so is real. The friction matrix is not symmetric in general, and so is complex, and is its complex conjugate, so that
In this context, the process is reversible if and only is symmetric, i.e., is real. The Sylvester equation (2.15) yields
We have thus
Let us add a remark on finite-ranged matrices. Let be a circulant matrix such that for only. The integer is dubbed the range of the matrix . As soon as the system size obeys , the amplitudes labelled by the discrete Fourier index are the restriction to the quantised momenta
of the trigonometric polynomial
which is independent of . When is symmetric, the above condition on the system size can be relaxed to . We shall see an example of such a situation with in section 3, devoted to the Gaussian spin model on a ring.
2.10 The case
To close this general section, we give explicit expressions of all quantities introduced so far in the case of two coupled degrees of freedom. This situation already exhibits most of the generic features of multivariate Ornstein-Uhlenbeck processes (see  for a recent account).
We parametrise the friction and diffusion matrices as
The friction matrix has positive eigenvalues for and , whereas the diffusion matrix is positive definite for and .
The reversibility condition (2.16) amounts to one single bilinear condition:
The solution of the Sylvester equation (2.15) reads
Let us introduce the irreversibility parameter (see (2.67))
Equation (2.18) yields
The entropy production rate reads
Equation (2.39) yields
We have therefore
In the present situation, there is one single reversibility condition (2.67), and therefore one single irreversibility parameter . As a consequence, all the quantities characterising the irreversibility are related to each other. We have indeed
3 The Gaussian spin model on a ring
Our first example of a multivariate Ornstein-Uhlenbeck process originates in a microscopic model, which is the one-dimensional version of the ferromagnetic Gaussian spin model . We shall endow the model with a dynamics where spatial asymmetry results in irreversibility. The main control parameters will be the system size and the asymmetry parameter . In this section we investigate the dynamics on a ring, whereas the more intricate situation of an open chain will be dealt with in section 4.
Let us begin with a brief report on the statics of the model. We consider the geometry of a finite ring of sites (), with periodic boundary conditions. Each site hosts a continuous spin . Throughout section 3, the index is understood mod . At infinite temperature, each spin has a Gaussian distribution such that . The weight of a spin configuration is therefore proportional to , where the free action reads
Interactions between spins are described by the nearest-neighbor ferromagnetic Hamiltonian
with . At temperature , the weight of a spin configuration is therefore proportional to , where the full action reads
The model owes its name to the fact that its Boltzmann weight is Gaussian. Its statics is characterised by the covariance matrix whose entries are . By definition, is the inverse of the symmetric matrix associated with the quadratic form (3.3). The non-zero elements of are and , with periodic boundary conditions. Translational invariance allows one to use the discrete Fourier transform introduced in section 2.9. As anticipated there, all the quantities labelled by the Fourier index are the restrictions of smooth functions of to the discrete set of momenta
In Fourier space, with the same conventions as in section 2.9, we have
and therefore , with
The range of is , and so the above expression holds for all .
The model is well-defined as long as the quadratic form (3.3) is positive definite. This condition amounts to requesting that all the eigenvalues , which appear as denominators in (3.7), are positive. The smallest of them, corresponding to , i.e., , reads . It vanishes at the critical point
The model therefore exists only in its high-temperature phase (). The existence of a non-zero critical temperature in one dimension underlines the lack of realism of the ferromagnetic Gaussian model . In spite of this, the dynamics of this model provides interesting examples within the framework of the present study.
In order to derive a closed-form expression of the correlation function , instead of evaluating the sum in (3.7), it is more convenient to observe that it obeys the difference equation
with periodic boundary conditions, expressing the identity . Looking for a solution of the form , we obtain after some algebra
The mass (inverse correlation length) is given by
and vanishes at the critical point according to
The correlation function diverges uniformly as
irrespective of the distance , as the critical point is approached.
In the thermodynamic limit (), keeping fixed, the expression (3.10) simplifies to
3.2 Gradient dynamics
The most natural dynamics for the Gaussian spin model is the gradient dynamics defined by
with periodic boundary conditions. The are independent white noises such that
This dynamics corresponds to an Ornstein-Uhlenbeck process with cyclic symmetry, as defined in section 2.9. The diffusion matrix is the unit matrix, whereas the friction matrix coincides with the matrix associated with the quadratic form (3.3). The proportionality between and is a common characteristic feature of linear gradient dynamics. In particular, the matrix is symmetric, and so the process is reversible. The expression of the covariance matrix (see (2.17)) coincides with the static result . We thus readily recover the static properties of the model recalled in section 3.1.
3.3 Asymmetric dynamics
We now deform the gradient dynamics (3.15) into the following asymmetric one:
with an arbitrary spatial asymmetry parameter (). This is the most general form keeping both the linearity and the range of dynamical interactions. The ferromagnetic spherical model with an asymmetric dynamics of this kind has been investigated in , and in more detail in . Ising spin models where spatially asymmetric rules induce irreversibility have been studied as well [6, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34].
The asymmetric dynamics (3.17) again corresponds to an Ornstein-Uhlenbeck process with cyclic symmetry, as defined in section 2.9. The diffusion matrix is still the unit matrix. The non-zero elements of the friction matrix are , and .
The irreversibility of the process is measured by the spatial asymmetry parameter : as soon as is non-zero, is not symmetric, and the process is irreversible. The remainder of this section 3 is devoted to characterising the nonequilibrium stationary state associated with the asymmetric dynamics (3.17) on a ring. In Fourier space, with the same conventions as in section 2.9, we have , whereas
The eigenvalues of are therefore complex. They lie on the ellipse centered at unity with semi-axes and . Their real parts do not depend on the asymmetry parameter . In particular, the spectral gap (see (2.6))
is independent of and vanishes as the critical point is approached.
Using (2.58), we obtain an expression of which coincides with (3.6), irrespective of . In other words, the stationary state is independent of the asymmetry parameter , i.e., of the irreversibility of the process. This property originates in the process being cyclically symmetric. The very same property was already observed in the spherical model with spatially asymmetric dynamics in the thermodynamic limit in any dimension . It is however not granted in general. It will indeed turn out to be violated in the geometry of an open chain (see section 4.2).
3.4 Thermodynamic limit
Within the formalism exposed in section 2.9, the expression (3.18) yields explicit results for intensive quantities characterising the nonequilibrium stationary state in the thermodynamic limit (), where sums over become integrals over .
As a first example, we consider the central moments of the spectrum of , i.e.,
In the thermodynamic limit, we have
Let us now turn to physical quantities. The entropy production rate is extensive, i.e., there is a well-defined intensive entropy production rate per unit time and per spin,
This quantity is proportional to , with an amplitude which remains finite at the critical point.
We have furthermore
The asymmetry index of the process (see (2.20)) is reached for and reads
This result is proportional to , with an amplitude which diverges as the critical point is approached.
Finally, the typical FDR reads (see (2.61))
This quantity exhibits a richer dependence on parameters. For , it departs from its equilibrium value as
For , i.e., when interactions are totally asymmetric, we have . For , falls off as
At the critical point, (3.26) simplifies to
3.5 Finite-size effects
Intensive quantities pertaining to finite-size samples generically converge exponentially fast to their thermodynamic limit, except near the critical point, where slow convergence properties and finite-size scaling laws can be expected. Let us take the example of the entropy production rate , which is the easiest to analyse from a technical viewpoint. Equations (2.63) and (3.18) yield for all
The sum can be performed exactly by means of the identity (see e.g. [56, Eq. (41.2.8)])
We obtain after some algebra the closed-form expression
For , the parameter drops out of the dynamical equations (3.17), so that the process is symmetric and reversible, and so vanishes.
All over the high-temperature phase, the difference falls off as . Right at the critical point, we have
with a non-vanishing finite-size correction, a rather unusual feature for a system with periodic boundary conditions. Finally, in the scaling region around the critical point, i.e., for large and small, the entropy production rate obeys a finite-size scaling law of the form
The product is the natural scaling variable of the problem, as it is the dimensionless ratio between the system size and the correlation length .
4 The Gaussian spin model on an open chain
In this section we investigate the dynamics of the ferromagnetic Gaussian spin model on an open chain.
Let us start with the statics of the model. We consider the geometry of a finite open chain of sites (), where each site hosts a continuous spin . At temperature , the weight of a spin configuration is still proportional to , with the full action given by (3.3), albeit with Dirichlet boundary conditions (). The matrix , whose entries are the correlation functions , is again the inverse of the symmetric matrix associated with the latter quadratic form. We thus have
with Dirichlet boundary conditions in both variables (). Looking for solutions of the form , separately for and for , we obtain after some algebra the expression
to be completed by symme