1 Introduction↩︎

In this review, we focus on the temporal behaviour of the Gibbs’ and conditional entropies in dynamical and semi-dynamical systems with both stochastic perturbations as well as delayed dynamics, building on and extending the recent results of [1] as well as older results [2], [3]

The outline of the paper is as follows. Section 2 defines the the Gibbs’ and conditional entropies, while Section 3 shows that these entropies are constant in time in invertible deterministic system (e.g. a system of ordinary differential equations). Section 4 introduces the dynamic concept of asymptotic stability,³, and two main results connecting the convergence of the conditional entropy with asymptotic stability (Theorem 2), and the existence of unique stationary densities with the convergence of the conditional entropy (Theorem 3). Section 5 considers the stochastic extension where a system of ordinary equations is perturbed by Gaussian white noise (thus becoming non-invertible) and gives general results showing that in this stochastic case asymptotic stability holds. In this section we give examples of the effects of noise on the behaviour of the Gibbs and conditional entropy. Following this we turn to an examination of the effects of delays in the dynamics on the evolution of the conditional entropy in Section 6. We conclude with a short discussion.

2 Entropies: Gibbs and conditional.↩︎

Two types of entropies are considered here. The first is the Gibbs’ entropy, which is an extension of the equilibrium entropy introduced by [4] to a time dependent situation. This has been considered by a number of authors, namely [5], [6], [7], [8] and [9]–[11].

In his seminal work [4], assuming the existence of a system steady state density \(f_*\) on the phase space \({ X}\), introduced the concept of the index of probability given by \((-\ln f_*(x) )\) where “\(\ln\)" denotes the natural logarithm. He then identified the entropy in a steady state situation with the average of the index of probability \[H_G(f_*) = - \int_{ X} f_*(x) \ln f_*(x)\, dx. \label{e-gibbs}\tag{1}\] This is called the equilibrium or steady state Gibbs’ entropy.

The Gibbs’ equilibrium entropy definition Eq. 1 has repeatedly proven to yield correct results when applied to a variety of equilibrium situations [12]–[17] and this is why it is the gold standard for equilibrium computations in statistical mechanics and thermodynamics. Thus it makes total sense to identify the equilibrium Gibbs’ entropy \({H_G(f_*)}\) with the equilibrium thermodynamic entropy.

The question of how a time dependent non-equilibrium entropy should be defined has interested investigators for some time, and specifically the question of whether the Gibbs’ entropy \(H_G(f)\) should be extended to a time dependent situation has occupied many researchers. Various aspects of this question have been considered [5], [6], [8], [18], [19] and if the definition of the steady state Gibbs’ entropy is extended to time dependent (non-equilibrium) situations then the time dependent Gibbs’ entropy of a density \(f(x,t)\) is defined by \[H_G(f) = - \int_{ X} f(x,t) \ln f(x,t)\, dx. \label{e-gibbstime}\tag{2}\]

A second type of entropy, the conditional entropy, is a generalization of the Gibbs’ entropy. Convergence properties of the conditional entropy have been extensively studied because entropy methods have been known for some time to be useful for problems involving questions related to convergence of solutions in partial differential equations [20]–[25]. Explicitly, we define the conditional entropy as [26] \[H_c(f|f_*)=-\int_X f(x,t)\ln\dfrac{f(x,t)}{f_*(x)}dx. \label{d:conent}\tag{3}\] This is also known as the Kullback-Leibler or relative entropy [20], or the relative Boltzmann entropy [27], and has been related to the thermodynamic free energy [28].

3 Gibbs’ and conditional entropy in invertible deterministic systems↩︎

To set the stage let \((X,\mathcal{B},\mu)\) be a \(\sigma\)-finite measure space. Further, let \(\{P^t\}_{t\ge 0}\) be a semigroup of Markov operators on \(L^1(X,\mu)\), i.e. \(P^tf\ge 0\) for \(f\ge 0\), \(\int P^tf(x)\, \mu(dx)=\int f(x) \, \mu(dx)\), and \(P^{t+s}f=P^t(P^sf)\). If the group property holds for \(t,s \in \mathbb{R}\), then we say that \(P\) is invertible, while if it holds only for \(t,s \in \mathbb{R}^+\) we say that \(P\) is non-invertible. We denote the corresponding set of densities by \(\mathcal{D}(X,\mu)\), or \(\mathcal{D}\) when there will be no ambiguity, so \(f \in \mathcal{D}\) means \(f \geq 0\) and \(||f||_1=\int_X f(x) \,\mu(dx) = 1\). A density \(f_*\) is called a stationary density of \(P^t\) if \(P^tf_*=f_*\) for all \(t\).

In this section we briefly consider entropy behaviour in situations where the dynamics are invertible in the sense that they can be run forward or backward in time without ambiguity. To make this clearer, consider a phase space \(X\) and a dynamics \(S_t:X \to X\). For every initial point \(x^0\), the sequence of successive points \(S_t(x^0)\), considered as a function of time \(t\), is called a trajectory. In the phase space \(X\), if the trajectory \(S_t(x^0)\) is nonintersecting with itself, or intersecting but periodic, then at any given final time \(t_f\) such that \(x^f = S_{t_f}(x^0)\) we could change the sign of time by replacing \(t \rightarrow (-t)\), and run the trajectory backward using \(x^f\) as a new initial point in \(X\). Then our new trajectory \(S_{-t}(x^f)\) would arrive precisely back at \(x^0\) after a time \(t_f\) had elapsed: \(x^0 = S_{-t_f}(x^f)\). Thus in this case we have a dynamics that may be reversed in time completely unambiguously.

We formalize this by introducing the concept of a dynamical system \(\lbrace S_t \rbrace _{t \in \mathbb{R}}\), which is simply any group of transformations \(S_t:X \rightarrow X\) having the two properties: 1. \(S_0 (x) = x\); and 2. \(S_t(S_{t'}(x)) = S_{t+t'}(x)\) for \(t,t'\in \mathbb{R}\) or \(\mathbb{Z}\). Since, from the definition, for any \(t \in \mathbb{R}\), we have \(S_t(S_{-t}(x)) = x = S_{-t}(S_t(x))\), it is clear that dynamical systems are invertible in the sense discussed above since they may be run either forward or backward in time. Systems of ordinary differential equations are examples of dynamical systems.

The first result is very general, and shows that the conditional entropy of any invertible system is constant and uniquely determined by the method of system preparation. Thus

In the case where we are considering a deterministic dynamics \(S_t: {\cal X} \to {\cal X}\) where \({\cal X} \subset X\), then the corresponding Markov operator is also known as the Frobenius Perron operator [26], and is given explicitly by \[\nonumber P^tf_0(x) = f_0(S_{-t}(x)) | J^{-t}(x)|\] where \(J^{-t}(x)\) denotes the Jacobian of \(S_{-t}(x)\). A simple calculation shows \[\begin{align} H_c(P^tf_0|f_*) &=& -\int_ {\cal X} P^t f_0(x)\log \left [ \dfrac{P^t f_0(x)}{f_*(x)}\right ]\, dx \nonumber \\ &=& -\int_ {\cal X} f_0(S_{-t}(x))|J^{-t}(x)|\log \left [\dfrac{ f_0(S_{-t}(x)) }{f_*(S_{-t}(x))}\right ]\, dx \nonumber \\ &=& -\int_ {\cal X} f_0(y)\log\left [ \dfrac{ f_0( y) } {f_*(y)}\right ] \, dy \nonumber \\ &\equiv& H_c(f_0|f_*) \nonumber \end{align}\] as expected from Theorem 1.

More specifically, if the dynamics corresponding to our invertible Markov operator are described by the system of ordinary differential equations \[\dfrac{dx_i}{dt} = F_i(x) \qquad i = 1,\ldots ,d \label{ode}\tag{4}\] operating in a region of \(\mathbb{R}^d\) with initial conditions \(x_i(0) = x_{i,0}\), then [26] the evolution of \(f(x,t) \equiv P^tf_0(x)\) is governed by the generalized Liouville equation \[\frac{\partial f}{\partial t} = -\sum_i \frac{\partial (fF_i)}{\partial x_i}. \label{e-leqn}\tag{5}\] The corresponding stationary density \(f_*\) is given by the solution of \[\sum_i \frac{\partial (f_* F_i)}{\partial x_i} = 0.\] Note that the uniform density \(f_* \equiv 1\), meaning that the flow defined by Eq. 4 preserves the Lebesque measure, is a stationary density of Eq. 5 if and only if \[\sum_i \frac{\partial F_i}{\partial x_i} = 0.\]

In particular, for the system of ordinary differential equations (4 ) whose density evolves according to the Liouville equation (5 ) we can assert that the conditional entropy of the density \(P^tf_0\) with respect to the stationary density \(f_*\) will be constant for all time and will have the value determined by the initial density \(f_0\) with which the system is prepared. This result can also be proved directly by noting that from the definition of the conditional entropy we may write \[H_c(f|f_*) = - \int_{\mathbb{R}^d} f(x) \left[ \log \left( \frac{f}{f_*} \right) + \frac{f_*}{f} - 1 \right] \, dx\] when the stationary density is \(f_*\). Differentiating with respect to time gives \[\frac{dH_c}{dt} = - \int_{\mathbb{R}^d} \frac{df}{dt} \log \left[ \frac{f}{f_*} \right] \,dx\] or, after substituting from (5 ) for \((\partial f/\partial t)\), and integrating by parts under the assumption that \(f\) has compact support, \[\frac{dH_c}{dt} = \int_{\mathbb{R}^d} \frac{f}{f_*} \sum_i \frac{\partial (f_*F_i)}{ \partial x_i} \,dx.\] However, since \(f_*\) is a stationary density of \(P^t\), it is clear from (5 ) that \[\frac{dH_c}{dt} = 0, \label{e:cond1}\tag{6}\] and we conclude that the conditional entropy \(H_c(P^tf_0|f_*)\) does not change from its initial value when the dynamics evolve in this manner. It is obvious that this conclusion also holds for the Gibbs’ entropy.

4 Asymptotic stability and entropy behaviour↩︎

We call a semigroup of Markov operators \(P^t\) on \(L^1(X,\mu)\) asymptotically stable if there is a density \(f_*\) such that \(P^tf_*=f_*\) for all \(t>0\) and for all densities \(f\) \[\lim_{t\to\infty}||P^tf-f_*||_1=0.\]

Thus from this theorem asymptotic stability is necessary and sufficient for the convergence of \(H_c\) to zero. Our next result draws a connection between the existence of a unique stationary density \(f_*\) of \(P^t\) and the convergence of the conditional entropy \(H_c\). Recall that the semigroup \(P^t\) on \(L^1(X,\mu)\) is called continuous if \(P^tf\) converges to \(f\) as \(t\downarrow 0\) for every density \(f\) and it is partially integral if there exists a measurable function \(k : X \times X \to [0,\infty)\) and \(t_0>0\) such that \[P^{t_0}f(x) \ge \int_X k(x, y)f(y)\,\mu(dy)\] for every density \(f\) and \[\int_X \int_X k(x, y)\,\mu(dy)\,\mu(dx)>0.\]

Since the conditional and Gibbs’ entropies are related by \[H_G(P^tf_0 )=H_c(P^tf_0 |f_*)-\int P^tf_0(x)\log f_*(x) \mu(dx), \label{d:con-bg-ent1}\tag{7}\] Theorem 2 implies the next result, see [3].

These results are very general in their statements about the behavior of the conditional and Gibbs’ entropies. Theorem 1 tells us that when the dynamics are such that \(P^t\) is a group (we have a dynamical system) the conditional entropy will be constant and fixed by the initial value of \(f_0\). However, when \(P^t\) is an asymptotically stable semigroup then Theorems 2 and 4 respectively guarantee the convergence of the conditional entropy to its maximal value of zero and the Gibbs’ entropy to its equilibrium value.

5 Effects of Gaussian noise↩︎

In thermodynamic considerations physicists often assume and discuss an idealized situation of a system isolated from the environment, and by that they mean that the system under consideration can exchange neither energy nor matter with its surroundings. This is, of course, an unachievable idealization. In reality any system is subjected to perturbations from multiple outside sources, and from the Central Limit Theorem one would expect that the summated perturbations would be at least approximately Gaussian distributed. Here we consider the effects of such a Gaussian distributed disturbance on dynamics and on entropies.

We consider the behaviour of the stochastically perturbed system \[\dfrac{dx_i}{dt} = F_i(x) + \sum_{j=1}^n \sigma_{ij}(x) \xi _j, \qquad i = 1,\ldots ,d \label{stochode}\tag{8}\] with the initial conditions \(x_i(0) = x_{i,0}\), where \(\sigma _{ij}(x)\) is the amplitude of the stochastic perturbation and \(\xi_j = \dfrac {dw_j}{dt}\) is a “white noise" term that is the derivative of a Wiener process \(w\). In matrix notation we can rewrite Eq. 8 as \[dx(t)=F(x(t))dt + \sigma(x(t)) \, dw(t), \label{stochodem}\tag{9}\] where \(\sigma(x)=[\sigma_{ij}(x)]\) and \(w\) is an \(n\)-dimensional Wiener process.⁴

The Fokker-Planck equation that governs the evolution of the density function \(f(x,t)\) of the process \(x(t)\) generated by the solution to the stochastic differential equation (9 ) is given by [33]–[35] \[\frac{\partial f}{\partial t} = - \sum_{i=1}^d \frac{\partial [F_i(x)f]}{\partial x_i} +\frac{1}{2} \sum_{i,j=1}^d \frac{\partial ^2 [a_{ij}(x)f]}{\partial x_i \partial x_j} \label{fpeqn}\tag{10}\] where \[a_{ij}(x)=\sum_{k=1}^{n}\sigma_{ik}(x)\sigma_{jk}(x).\]

If \(p(x,t|x_0)\) is the probability density function of \(x(t)\) conditional on \(x(0)=x_0\) then \(p(x,t|x_0)\) is the fundamental solution of the Fokker-Planck equation, i.e. for every \(x_0\) the function \((x,t)\mapsto p(x,t|x_0)\) is a solution of the Fokker-Planck equation with the initial condition \(\delta(x-x_0)\). The general solution \(f(x,t)\) of the Fokker-Planck equation (10 ) with the initial condition \[f(x,0)=f_0(x)\] is then given by \[f(x,t)=\int p(x,t|x_0)f_0(x_0)\, dx_0. \label{gensoln}\tag{11}\] Define the Markov operators \(P^t\) by \[P^tf_0(x)=\int p(x,t|x_0)f_0(x_0)\, dx_0, \quad f_0\in L^1. \label{mo}\tag{12}\] Then \(P^tf_0\) is the density of the solution \(x(t)\) of Eq. 9 provided that \(f_0\) is the density of \(x(0)\).

If the coefficients \(a_{ij}\) and \(F_i\) are sufficiently regular then a fundamental solution \(p\) exists. One such set of conditions is the following: (1) the \(F_i\) are of class \(C^2\) with bounded derivatives; (2) the \(a_{ij}\) are of class \(C^3\) and bounded with all derivatives bounded; and (3) the uniform parabolicity condition holds, i.e. there exists a strictly positive constant \(\rho>0\) such that \[\sum_{i,j=1}^da_{ij}(x)\lambda_i\lambda_j\ge \rho \sum_{i=1}^d \lambda_i^2,\qquad \lambda_i,\lambda_j\in \mathbb{R}, x\in \mathbb{R}^d.\] The uniform parabolicity condition implies that \(p(t,x|x_0)>0\), and thus \(P^tf(x)>0\) for every density, which implies that there can be at most one stationary density, and that if it exists then \(f_*>0\). In this setting, asymptotic stability holds if and only if there is a stationary density \(f_*\) that satisfies (13 ).

Gibb’s entropy↩︎

The steady state density \(f_*(x)\) is the stationary solution of the Fokker Planck Eq. (10 ): \[- \sum_{i=1}^d \frac{\partial [F_i(x)f]}{\partial x_i} +\frac{1}{2} \sum_{i,j=1}^d \frac{\partial ^2 [a_{ij}(x)f]}{\partial x_i \partial x_j} = 0. \label{ssfpeqn}\tag{13}\] Differentiating Eq. 2 for the Gibbs’ entropy yields \[\dfrac{dH_{G}}{dt} = \int f \left(\sum_i \frac{\partial F_i(x)}{\partial x_i} - \dfrac12 \sum_{i,j} \frac{\partial^2 a_{ij}(x)}{\partial x_i \partial x_j}\right) \,dx + \dfrac12 \int \dfrac{1}{f} \sum_{i,j=1}^d a_{ij}(x)\dfrac {\partial f}{\partial x_i} \dfrac {\partial f}{\partial x_j}\, dx. \label{bgnoise}\tag{14}\] If the \(a_{ij}\) are independent of \(x\) then we obtain \[\nonumber \dfrac{dH_{G}}{dt} = \int f \sum_i \frac{\partial F_i(x)}{\partial x_i} \,dx + \dfrac12 \sum_{i,j=1}^d a_{ij}\int \dfrac{1}{f} \dfrac {\partial f}{\partial x_i} \dfrac {\partial f}{\partial x_j}\, dx. \label{c-bgnoise}\tag{15}\] The first term is of indeterminant sign [8], but the second is positive definite so the temporal behavior of the Gibbs’ entropy in the presence of Gaussian noise is unclear.

Conditional entropy↩︎

Differentiating Eq. 3 for the conditional entropy with respect to time, and using Eq. 10 with integration by parts, along with the fact that since \(f_*\) is a stationary density it satisfies (13 ), we obtain \[\nonumber \dfrac{dH_c}{dt} = \dfrac12 \int \left( \dfrac{f_*^2}{f} \right) \sum_{i,j=1}^{d} a_{ij}(x) \dfrac {\partial}{\partial x_i} \left( \dfrac {f}{f_*} \right) \dfrac {\partial}{\partial x_j} \left( \dfrac{f}{f_*} \right) \,dx. \label{dtcondent}\tag{16}\] Since the matrix \((a_{ij}(x))\) is nonnegative definite, one concludes that \[\dfrac{dH_c}{dt}\ge 0. \label{e:cond2}\tag{17}\] This implies that the conditional entropy is a monotonic function of time.

We now discuss the speed of convergence to zero of the conditional entropy. It is known that under some conditions [23], [36] there exists a constant \(\lambda>0\) such \[H_c(P^tf_0|f_*)\ge e^{-2\lambda t}H_c(f_0|f_*)\] for all initial densities \(f_0\) with \(H_c(f_0|f_*)>-\infty\). See [2] for a review of these conditions. Here we use a different approach.

Note that for \(P^t\) as in 12 we have the following lower bound for the conditional entropy \[\label{LB} H_c(P^tf_0|f_*)\ge \int H_c(p(\cdot,t|x_0)|f_*)f_0(x_0)dx_0\tag{18}\] for any initial density \(f_0\). To see this use Jensen’s inequality to obtain \[\eta\left(\frac{P^tf_0(x)}{f_*(x)}\right)=\eta\left(\int \frac{p(x,t|x_0)}{f_*(x)}f_0(x_0)dx_0\right)\ge \int \eta\left(\frac{p(x,t|x_0)}{f_*(x)}\right)f_0(x_0)dx_0,\] where \(\eta\) is the concave function \(\eta(s)=-s\log s\) for \(s>0\). Next multiply by \(f_*(x)\), integrate with respect to \(x\) and change the order of integration to obtain the lower bound.

5.1 Example of an Ornstein-Uhlenbeck process↩︎

Consider the scalar linear differential equation with additive noise \[\label{1dOU} dx(t)=ax(t)dt+\sigma dw(t),\tag{19}\] where \(a,\sigma\in \mathbb{R}\) and \(\sigma>0\). If we take \(a=-\gamma\) with \(\gamma>0\) then Eq. 19 defines the Ornstein-Uhlenbeck process, which was historically developed in thinking about perturbations to the velocity of a Brownian particle. The solution \(x(t)\) of 19 is of the form \[x(t)=e^{ta}x(0)+\sigma\int_0^te^{(t-r)a}dw(r).\] Note that \[\eta(t)=\sigma\int_0^te^{(t-r)a}dw(r),\quad t\ge 0,\] is a Gaussian process with mean zero and covariance of the form \[\mathbb{E}(\eta(t)\eta(t+s))=\sigma^2\int_0^t e^{(t-r)a}e^{(t+s-r)a}dr=\sigma^2 \int_0^t e^{ra}e^{(r+s)a}dr,\quad t,s\ge 0,\] by the Itô isometry. If \(x(0)=x_0\) then the random variable \(x(t)\) has a Gaussian distribution with mean \(\mathbb{E}(x(t))=e^{ta}x_0\) and variance \[\textrm{Var}(x(t))=\mathbb{E}([x(t)-\mathbb{E}(x(t))]^2)=\mathbb{E}(\eta(t)^2)=\sigma^2\int_0^t[e^{ra}]^2dr.\] Thus the probability density function of \(x(t)\) conditional on \(x(0)=x_0\) is given by \[p(x,t|x_0)=\frac{1}{\sqrt{2 \pi \upsilon(t)}}\exp \left \{-\frac{(x-e^{ ta}x_0)^2}{2\upsilon(t)}\right \}, \label{o-utdep}\tag{20}\] where \[\upsilon(t):=\sigma^2\int_0^t[e^{ra}]^2dr.\] If \(x(0)\) has a Gaussian distribution with mean \(0\) and variance \(\sigma_*^2\) then \(x(t)\) is again Gaussian with mean \(0\) and variance \[\sigma_t^2=e^{2at}\sigma_*^2+\upsilon(t).\] Note that \(\sigma_t^2\) is independent of \(t\) if and only if \[\label{exOU} a<0\quad \text{and}\quad \sigma_*^2=-\frac{\sigma^2}{2a}.\tag{21}\] If condition 21 holds then we define the process \[Y(t)=\sigma\int_{-\infty}^{t}e^{(t-r)a}dw(r),\quad t\in \mathbb{R},\] where \(w\) is extended to \(\mathbb{R}\) by taking an independent copy \(\{\overline{w}(t):t\ge 0\}\) of the Wiener process and setting \(w(t)=-\overline{w}(-t)\) for \(t<0\). Observe that \[Y(t)=e^{ta}Y(0)+\eta(t),\quad t\ge 0.\] Thus \(Y(t)\) is a solution of 19 and \(Y(t)\) has a Gaussian distribution with mean 0 and variance \(\sigma_*^2\). Consequently, \(Y(t)\), \(t\ge 0\), is a stationary solution of 19 . Its covariance function is of the form \[\mathbb{E}(Y(s)Y(t+s))=\sigma_*^2e^{ta}, \quad t\ge 0.\] Thus, if \(a<0\) then \[\sigma_*^2=\lim_{t\to\infty}\sigma^2\int_0^t[e^{ra}]^2dr=\sigma^2\int_0^\infty[e^{ra}]^2dr<\infty\] is the only solution of \[\label{e:ibv0} 2a\sigma_*^2+\sigma^2=0.\tag{22}\] We have \[\lim_{t\to \infty}p(x,t|x_0)=f_*(x),\] where \[f_*(x)=\frac{1}{\sqrt{2 \pi \sigma_*^2}}\exp \left \{-\frac{x^2}{2\sigma_*^2}\right \}\] is the density of \(Y(t)\). The function \(f_*\) is then the unique stationary solution to the corresponding Fokker-Planck equation \[\nonumber \dfrac{\partial f}{\partial t} = -\dfrac {\partial [a x f]}{\partial x} + \dfrac {\sigma^2}2 \dfrac {\partial^2 f}{\partial x^2}. \label{o-ufpeqn}\tag{23}\]

We may now examine how the two different types of entropy behave.

Gibb’s entropy↩︎

It is easily seen that the Gibbs’ entropy of the Gaussian density \[f_{m,\sigma^2}(x)=\frac{1}{\sqrt{2\pi \sigma^2}}\exp{\left\{-\frac{1}{2\sigma^2}(x-m)^2\right\}}, \quad x\in \mathbb{R},\] where \(\sigma>0\) and \(m\in \mathbb{R}\), is \[H_G(f_{m,\sigma^2}) = -\int _{-\infty}^{+\infty}f_{m,\sigma^2}(x) \ln f_{m,\sigma^2}(x) dx = \dfrac 12 + \dfrac 12\ln (2 \pi \sigma^2). \label{e:gibbs-1d}\tag{24}\]

It is not the case that the Gibbs’ entropy will always be an increasing function of time when the dynamics are perturbed by an Ornstein-Uhlenbeck process. To see this consider as the initial density \(f_0\) a Gaussian \[f_0(x) = \dfrac {1}{ \sqrt{2 \pi\bar \sigma^2}} \exp{\left \{ -\dfrac {x^2}{2 \bar \sigma^2} \right \}}\nonumber\] where \(\bar \sigma >0\), then \[P^tf_0 (x) = \dfrac {1}{ \sqrt{2 \pi\bar{\sigma}_t^2}} \exp \left \{ - \dfrac {x ^2}{2 \bar{\sigma}_t^2} \right \}\nonumber\] where \[\bar{\sigma}^2_t = \sigma_*^2 + (\bar \sigma^2 - \sigma_*^2)e^{2a t}\nonumber\] with \(\sigma_*^2 = -\sigma^2/2a\) and \(a<0\). The Gibbs’ entropy is \[H_{G}(P^tf_0 ) = \dfrac 12+ \dfrac 12\ln (2 \pi\bar{\sigma}_t^2) \nonumber\] and \[\label{e:gibbs} \dfrac {dH_{G}(P^tf_0 )}{dt} \left \{ \begin{array}{lll} &>& 0 \qquad for \qquad \bar \sigma^2 < \sigma_*^2 \\ &=& 0 \qquad for \qquad \bar \sigma^2 = \sigma_*^2\\ &<& 0 \qquad for \qquad \bar \sigma^2 > \sigma_*^2 , \end{array} \right.\tag{25}\] implying that the evolution of the Gibbs’ entropy in time is a function of the statistical properties (\(\bar \sigma ^2\)) of the initial ensemble.

Note also that the Gibbs’ entropy of the transition probability \(p\) is an increasing function of time, since \[H_{G}(p(\cdot,t|x_0)) = \dfrac 12 +\dfrac 12\ln (2 \pi \upsilon(t))\] and the variance \(\upsilon(t)\) is a strictly increasing function.

Conditional entropy↩︎

Next, let us calculate the conditional entropy of two Gaussian densities \(f_{m_1,\sigma_1^2}\) and \(f_{m_2,\sigma_2^2}\), where \(\sigma_1,\sigma_2>0\) and \(m_1,m_2\in \mathbb{R}\). We have \[\ln \dfrac{f_{m_1,\sigma_1^2}(x)}{f_{m_2,\sigma_2^2}(x)}=\ln \sqrt{\dfrac{\sigma_2^2}{\sigma_1^2}}-\frac{1}{2\sigma_1^2}(x-m_1)^2+\frac{1}{2\sigma_2^2}(x-m_2)^2.\] Since \[\int_{\mathbb{R}} f_{m_1,\sigma_1^2}(x)x^2\, dx=\sigma_1^2+m_1^2\quadand\quad \int_{\mathbb{R}}f_{m_1,\sigma_1^2}(x)x\, dx=m_1,\] we arrive at \[H_c(f_{m_1,\sigma_1^2}|f_{m_2,\sigma_2^2})=\dfrac 12 \ln \dfrac{\sigma_1^2}{\sigma_2^2}+\dfrac{1}{2}\left(1-\dfrac{\sigma_1^2}{\sigma_2^2}\right)-\dfrac{1}{2\sigma_2^2}(m_1-m_2)^2. \label{e:conent1dg}\tag{26}\]

To obtain the speed of convergence of the conditional entropy to zero we use the lower bound from 18 . Observe that if we calculate the conditional entropy for the transition density \(p\) as in 20 then \[\label{ent:OU1d} H_c(p(\cdot,t|x_0)|f_*)=\frac{1}{2}\ln \frac{\upsilon(t)}{\sigma_*^2}+\frac{1}{2}\left(1-\frac{\upsilon(t)}{\sigma_*^2}\right)-\frac{1}{2\sigma_*^2}(e^{ta }x_0)^2,\tag{27}\] where \(\sigma_*^2=\lim_{t\to\infty} \upsilon(t)\). Consequently, we have for the one dimensional linear SDE with additive noise \[H_c(P^tf_0|f_*)\ge \frac{1}{2}\ln \frac{\upsilon(t)}{\sigma_*^2}+\frac{1}{2}\left(1-\frac{\upsilon(t)}{\sigma_*^2}\right)-\frac{1}{2\sigma_*^2}\int_{\mathbb{R}} (e^{ta }x_0)^2 f_0(x_0)dx_0.\] Since \[\lim_{t\to\infty} \upsilon(t)=\sigma_*^2\quad \text{and}\quad \lim_{t\to\infty}e^{ta }x_0=0\] we obtain \(\lim_{t \to \infty}H_c(P^tf_0|f_*) = 0\) in addition to the conclusion in 17 that \(\dot{H}_c(P^tf_0|f_*) \geq 0\).

6 The effects of delays↩︎

6.1 A Fokker-Planck-like formulation↩︎

We start with a few preliminaries. Throughout, \(C=C([-\tau,0],\mathbb{R}^d)\) denotes the Banach space of all continuous functions \(\phi\colon[-\tau,0]\to\mathbb{R}^d\) equipped with the supremum norm and the Borel \(\sigma\)-algebra. We initially focus on the equation without noise \[\label{e:dde} \begin{align} dx(t)=&\mathcal{F}(x(t),x(t-\tau))dt, \quad t\ge 0, \\ x(t)=&\phi(t),\quad t\in[-\tau,0], \end{align}\tag{28}\] where the initial function \(\phi\in C\) and \(\mathcal{F}\) are such that for each \(\phi\in C\) there exists a continuous function \(x\colon[-\tau,\infty)\to \mathbb{R}^d\) so 28 has a unique global solution that depends continuously on \(\phi\). For each \(t\ge 0\) define the solution map \(S_t\colon C\to C\) by \[\label{d:sol} S_t(\phi)(s)=x_t(s)=x(t+s)\quad \text{for } s\in[-\tau,0],\tag{29}\] where \(x(t)\) is a solution of 28 with \(x_0=\phi\). The transformation \((t,\phi)\mapsto S_t(\phi)\) is continuous and \(\{S_t\}_{t\in\mathbb{R}^+}\) is a semi-dynamical system on \(C\) [37].

Further, let \(B(C)\) be the space of bounded Borel measurable functions \(\psi \colon C\to \mathbb{R}\) with the supremum norm \[\|\psi \|_{\infty}=\sup_{\phi\in C}|\psi (\phi)|,\] where \(|\cdot|\) is a norm on \(\mathbb{R}^d\), and \(\mathcal{M}(C)\) (resp. \(\mathcal{M}_1(C)\)) denote the space of finite (resp. probability) Borel measures on \(C\). For any \(\psi \in B(C)\) and \(\mu\in \mathcal{M}(C)\) we use the customary scalar product notation \[\langle \psi ,\mu\rangle=\int_C \psi (\phi)\mu(d\phi).\] \(C_b\subset B(C)\) denotes the Banach space of all bounded uniformly continuous functions \(\psi\colon C\to \mathbb{R}\) with the supremum norm. Following [38] and [39], \(\psi_t\in C_b\), \(t>0\) is said to converge weakly to \(\psi\in C_b\) as \(t\to 0^{+}\) (denoted by \(\mathrm{w-}\lim_{t\to 0}\psi_t=\psi\)) if \[\lim_{t\to 0}\langle \psi_t,\mu\rangle=\langle \psi,\mu\rangle\quad \text{for each } \mu\in \mathcal{M}(C).\] This is equivalent to \(\lim_{t\to 0}\psi_t(\phi)=\psi(\phi)\) for every \(\phi\in C\), and \(\sup_t\left\Vert\psi_t\right\Vert_{\infty}<\infty\).

A semigroup \(\{T^t\}_{t\ge 0}\) of linear operators on the space \(C_b\) is weakly continuous at \(0\) if \(\mathrm{w-}\lim_{t\to 0}T^t\psi=\psi\) for all \(\psi\in C_b.\) The weak generator \(\mathcal{L}\colon \mathcal{D}(\mathcal{L})\subset C_b\to C_b\) of the semigroup \(\{T^t\}_{t\ge 0}\) is defined by [38], [39] \[\begin{align} \mathcal{D}(\mathcal{L})&=\{\psi\in C_b: \mathrm{w-}\lim_{t\to 0}\frac{1}{t}\bigl(T^t\psi-\psi\bigr)\; \text{exists}\},\\ \mathcal{L}\psi &=\mathrm{w-}\lim_{t\to 0}\frac{1}{t}\bigl(T^t\psi-\psi\bigr). \end{align}\] For \(\psi \in \mathcal{D}(\mathcal{L})\) and \(\mu \in \mathcal{M}(C)\) we have \[\label{e:weak} \langle T^t \psi,\mu\rangle=\langle \psi,\mu\rangle +\int_0^t \langle T^r(\mathcal{L}\psi),\mu\rangle dr,\quad t>0.\tag{30}\]

We next look at the version of 28 with a perturbation: \[\label{e:sdde} \begin{align} dx(t)=&\mathcal{F}(x(t),x(t-\tau))dt+\sigma(x(t),x(t-\tau))dw(t), \quad t\ge 0, \\ x(t)=&\phi(t),\quad t\in[-\tau,0], \end{align}\tag{31}\] where \(\{w(t)\}_{t\ge 0}\) is a stochastic perturbation, assumed to be a Wiener process, with an amplitude \(\sigma\), potentially dependent on \(x(t)\) as well as \(x(t-\tau)\). Assume that a pathwise solution \(x(t)\) of 31 exists, and let \[x_t(s)=x(t+s), \quad s\in [-\tau,0], t\ge 0.\] It is known [39] that \(\{x_t\}_{t\ge 0}\) is a \(C\)-valued Markov process with transition semigroup \[T^t\psi(\phi)=\mathbb{E}(\psi(x_t)|x_0=\phi).\] The semigroup \(\{T^t\}_{t\ge 0}\) is weakly continuous. Let \(\mu_0\in \mathcal{M}_1(C)\), and for each \(t\ge 0\) define the probability measure \(\mu_t\) on the space \(C\) as the distribution of \(x_t\), so \[\langle \psi,\mu_t\rangle =\langle T^t \psi,\mu_0\rangle,\quad t\ge 0, \psi\in C_b.\] It thus follows from 30 and a change of variables that \[\label{e:weakf} \langle \psi,\mu_t\rangle=\langle \psi,\mu_0\rangle +\int_0^t \langle \mathcal{L} \psi,\mu_r\rangle dr\tag{32}\] for all \(t>0\) and \(\psi \in \mathcal{D}(\mathcal{L})\). However, it is difficult (see [39]) to identify the domain \(\mathcal{D}(\mathcal{L})\) of the weak generator \(\mathcal{L}\), and so we introduce an extended generator for the process \(x_t\). This is defined as a linear operator \(\mathcal{L}\) from its domain \(\mathcal{D}\) to the set of all Borel measurable functions on \(C\), where \(\psi \in \mathcal{D}\) if for each \(t>0\) we have \[\int_0^t \langle |\mathcal{L} \psi|,\mu_r\rangle dr <\infty\] and 32 holds. Then \(\{\mu_t\}_{t\ge 0}\) is the solution of the equation \[\label{eq:infdim} \frac{\partial}{\partial t}\mu_t=\mathcal{L}^*\mu_t.\tag{33}\] We will use below a subset of \(\mathcal{D}\) that will allow us to change 33 into a partial differential equation 35 , see [40] for the general study of path-distribution dependent stochastic differential equations.

Let \(C_c^2(\mathbb{R}^d)\) denote the space of twice continuously differentiable functions with compact support, and consider the differential operator from the space \(C_c^2(\mathbb{R}^d)\) to the set of all Borel measurable functions on \(C\) defined by \[Lg(\phi)=\sum_{i=1}^d \mathcal{F}_i(\phi(0),\phi(-\tau))\frac{\partial g}{\partial x_i} (\phi(0))+\frac{1}{2}\sum_{i,j=1}^d a_{ij}(\phi(0),\phi(-\tau))\frac{\partial^2 g}{\partial x_i\partial x_j} (\phi(0)),\] for \(\phi\in C\), \(g\in C_c^2(\mathbb{R}^d).\) Here, \(a\) is the matrix \(\sigma\sigma^T\), where \(\sigma^T\) is the transpose of the matrix \(\sigma\). Let \(\mu(t)\) be the marginal distribution of the measure \(\mu_t\), defined on \(\mathbb{R}^d\) by \[\mu(t)(B)=\mu_t\{\phi\in C: \phi(0)\in B\}, \quad B\in \mathcal{B}(\mathbb{R}^d).\] It can be rewritten with the projection map \(\pi_0\colon C\to \mathbb{R}^d\) defined by \(\pi_0(\phi)=\phi(0)\), \(\phi\in C\), as \(\mu(t)=\mu_t\circ \pi_0^{-1}\). Thus \(\mu(t)\) is the distribution of \(x(t)\) for all \(t\ge 0\). Then we say that \(\{\mu_t\}_{t\ge 0}\) is a solution of the equation \[\frac{\partial}{\partial t}\mu(t)=L^*\mu_t\] if for each \(t> 0\) and \(g\in C_c^2(\mathbb{R}^d)\) \[\int_{0}^t \langle |Lg|,\mu_r\rangle dr<\infty\] holds, and \[\label{e:gLd} \int_{\mathbb{R}^d} g(x)\mu(t)(dx)=\int_{\mathbb{R}^d} g(x)\mu(0)(dx)+\int_{0}^t \langle Lg,\mu_r\rangle dr.\tag{34}\] Realize that this is equivalent to requiring \(\psi\in \mathcal{D}\) and \(\mathcal{L}\psi=Lg\) for \(\psi=g\circ \pi_0\) and all \(g\in C^2_c(\mathbb{R}^d)\). Furthermore, if we introduce the measure \[\nu(t)=\mu_t\circ \pi_{0,-\tau}^{-1},\quad t\ge 0,\] where \(\pi_{0,-\tau}\colon C\to \mathbb{R}^d\times \mathbb{R}^d\) is the projection map \(\pi_{0,-\tau}(\phi)=(\phi(0),\phi(-\tau))\), \(\phi\in C\), then \[\langle Lg,\mu_t\rangle=\int_{\mathbb{R}^d}\int_{\mathbb{R}^d}\left(\sum_{i=1}^d \mathcal{F}_i(x,y)\frac{\partial g}{\partial x_i} (x)+\frac{1}{2}\sum_{i,j=1}^d a_{ij}(x,y)\frac{\partial^2 g}{\partial x_i\partial x_j}(x)\right) \nu(t)(dx,dy),\] by the change of variables formula. The measure \(\nu(t)\) is the distribution of \((x(t),x(t-\tau))\).

Now suppose, additionally, that the measure \(\nu(t)\) has a density \(f_\nu\) with respect to the Lebesgue measure on \(\mathbb{R}^{2d}\) i.e., \(\nu(t)(dx,dy)=f_\nu(x, y,t)dxdy\). Then the measure \(\mu(t)\) has a density \(f\) with respect to the Lebesgue measure on \(\mathbb{R}^d\) and \[f(x,t)=\int_{\mathbb{R}^d}f_\nu(x,y,t)dy.\] We also have \[f(y,t-\tau)=\int_{\mathbb{R}^d}f_\nu(x,y,t)dx, \quad t\ge \tau.\] Therefore, we can rewrite 34 as \[\begin{align} &\int_{\mathbb{R}^d} g(x)f(x,t)dx=\int_{\mathbb{R}^d} g(x)f(x,0)dx\\ &+\int_0^t \int_{\mathbb{R}^d}\int_{\mathbb{R}^d} \left(\sum_{i=1}^d \mathcal{F}_i(x,y)\frac{\partial g}{\partial x_i} (x)+\frac{1}{2}\sum_{i,j=1}^d a_{ij}(x,y)\frac{\partial^2 g}{\partial x_i\partial x_j}(x)\right)f_\nu(x,y,r)dxdy dr, \end{align}\] which is the weak form of the evolution equation \[\label{e:weakL} \boxed{ \frac{\partial}{\partial t} f(x,t)=-\sum_{i=1}^d \frac{\partial}{\partial x_i}\int_{\mathbb{R}^d} \mathcal{F}_i(x,y)f_\nu(x,y,t)dy +\frac{1}{2}\sum_{i,j=1}^d \frac{\partial^2}{\partial x_i\partial x_j}\int_{\mathbb{R}^d}a_{ij}(x,y)f_\nu(x,y,t)dy. }\tag{35}\]

Note that Eq. 35 , which is our main result of this section, reduces to the Fokker-Planck equation if \(\mathcal{F}\) and \(a\) are independent of \(y\).

To obtain a lower bound on the conditional entropy for the case of a stochastic differential equation with delays, we make use of the same idea as in Section 5. Suppose that \(p(x,t|\phi)\) is the probability density function of the solution \(x(t)=x_t(0)\) of 31 with the initial condition being a deterministic function \(\phi\). By extending Eq. 11 to this case, we define \[\label{pdfd} f(x,t)=\int_C p(x,t|\phi)\mu_0(d\phi),\tag{36}\] where \(\mu_0\) is the distribution of the initial condition \(x_0=\phi\), which is now an element of the function space \(C\). Then we obtain the lower bound on the conditional entropy \[\label{Hcpdfd} H_c(f(\cdot,t)|f_*)\ge \int_C H_c(p(\cdot,t|\phi)|f_*)\mu_0(d\phi)\tag{37}\] as in Eq. 18 .

6.2 A linear first order example↩︎

Consider the linear scalar differential delay equation \[\label{e:lin} \begin{align} x'(t)&=ax(t)+bx(t-\tau),\quad t>0,\\ x(t)&=\phi(t),\quad t\in [-\tau,0], \end{align}\tag{38}\] where \(a,b\) are real constants and \(\phi \colon [-\tau,0]\to \mathbb{R}\) is a continuous function. We can write the solution of 38 as (see [37]) \[x(t)=X(t)\phi(0)+b\int_{-\tau}^0 X(t-r-\tau)\phi(r)dr,\quad t\ge 0,\] where \(X(t)\) is the fundamental solution of 38 , which means that \(X(t)\) satisfies 38 with \(X(t)=0\) for \(t<0\) and \(X(0)=1\). We have \[\label{e:funs} X(t)=\sum_{k=0}^{\lfloor t/\tau\rfloor} e^{a(t-k\tau)} \frac{b^k}{k!}(t-k\tau)^k,\quad t\ge 0,\tag{39}\] where \(\lfloor s\rfloor=\max\{k\in \mathbb{Z}: k\le s\}\). Thus for the solution map \(S_t\) we obtain \[S_t\phi(s)=X(t+s)\phi(0)+b\int_{-\tau}^0 X(t+s-q-\tau)\phi(q)dq,\quad t+s\ge 0, s\in [-\tau,0],t\ge0,\] and \(s_t\phi(s)=\phi(t+s)\) for \(t+s< 0\), \(s\in [-\tau,0]\), \(t\ge 0\).

Now consider the extension of 38 to a linear stochastic differential delay equation with additive noise \[\label{e:lins} dx(t)=(ax(t)+bx(t-\tau))dt+\sigma dw(t),\quad t\ge0,\tag{40}\] and the initial condition \[x(t)=\phi(t), \quad t\in [-\tau,0].\] Then \[x(t)=X(t)\phi(0)+b\int_{-\tau}^0 X(t-q-\tau)\phi(q)dq+\sigma\int_0^t X(t-q)dw(q),\] where \(X\) is the fundamental solution 39 . Note that \[y(t)=\sigma\int_0^t X(t-q)dw(q),\quad t\ge 0,\] is a Gaussian process with mean zero and covariance of the form \[\mathbb{E}(y(t)y(t+s))=\sigma^2\int_0^t X(t-q)X(t+s-q)dq=\sigma^2 \int_0^t X(q)X(q+s)dq,\quad t,s\ge 0,\] by Itô’s isometry. We thus have the following representation of the process \(x_t\) \[x_t=S_t\phi +y_t,\] where \(S_t\) is the solution map of the deterministic part 38 and \(y_t\) is defined by \[y_t(s)=\left\{ \begin{array}{ll} y(t+s), & ifs+t\ge 0,\\ 0, & if s+t<0. \end{array} \right.\] Since \[x_t(s)=S_t\phi(s)+y_t(s), \quad s\in [-\tau,0],\] and \(y_t(s)\) has mean zero, we obtain \(\mathbb{E}(x_t(s))=S_t\phi(s)\) and \[\mathrm{cov}(x_t(s_1),x_t(s_2))=\mathbb{E}(y_t(s_1)y_t(s_2))=\sigma^2 \int_0^{t+s_1} X(q)X(q+s_2-s_1)dq\] for \(-\tau \le s_1\le s_2\le 0\) and \(t+s_1>0\). Consequently, \(x_t\) is a Gaussian process with mean \(m_t\colon [-\tau,0]\to \mathbb{R}\) given by \[m_t(s)=S_t\phi(s), \quad s\in [-\tau,0],\] and the covariance function \(r_t\colon [-\tau,0]\times[-\tau,0]\to \mathbb{R}\) satisfies \[r_t(s_1,s_2)=\left\{ \begin{array}{ll} 0, & t+\min\{s_1,s_2\}\le 0, \\ \sigma^2 \int_0^{t+\min\{s_1,s_2\}} X(q)X(q+|s_2-s_1|)dq , & t+\min\{s_1,s_2\}> 0. \end{array} \right.\]

Existence of densities for the linear first order example↩︎

We check whether the random variable \(x(t)\) and the vector \((x(t),x(t-\tau))^T\) have densities when the initial condition is \(x(t)=\phi(t)\) for \(t\in [-\tau,0]\). The measure \(\mu(t)\), the distribution of \(x(t)=x_t(0)\), is Gaussian and has mean \(m_t(0)=S_t\phi(0)\) and variance \(\upsilon(t)=r_t(0,0)\). The variance \(\upsilon(t)= r_t(0,0)\) is always positive for \(t>0\), since \[\label{eq:varp} \upsilon(t)=\sigma^2\int_{0}^t X^2(s)ds\ge \sigma^2\int_{0}^{\min\{t,\tau\}} e^{as}ds>0.\tag{41}\] Hence the density \(p\) of \(\mu(t)\) is \[\label{e:densf} p(x,t)=\frac{1}{\sqrt{2\pi\upsilon(t)}}\exp \left\{-\frac{[x-S_t\phi(0)]^2}{2\upsilon(t)}\right\}.\tag{42}\] We introduce the conditional probability density of \(x_t(0)=x(t)\) conditional on \(x_0=\phi\) as \[p(x,t|\phi):=p(x,t).\] Further, the measure \(\nu(t)\), which is the distribution of the vector \((x(t),x(t-\tau))^T=(x_t(0),x_t(-\tau))^T\), is a two-dimensional Gaussian with mean vector \((m_t(0),m_t(-\tau))^T=(S_t\phi(0),S_t\phi(-\tau))^T\) and covariance matrix \[\label{e:Qt} Q_t=\left( \begin{array}{cc} r_t(0,0) & r_t(0,-\tau) \\ r_t(-\tau,0) & r_t(-\tau,-\tau) \\ \end{array} \right).\tag{43}\] Notice that \(r_t(-\tau,-\tau)=0\) and \(r_t(-\tau,0)=0\) for \(t\le \tau\). Hence, if \(t\le \tau\) then \(\det (Q_t)=0\) and \(\nu(t)\) does not have a density.

We now show that \(\nu(t)\) has a density for \(t>\tau\). To see this observe that \[r_t(-\tau,0)=\sigma^2\int_0^{t-\tau} X(s)X(s+\tau)ds,\] and thus \[r_t(-\tau,0)^2\le \sigma^4\int_0^{t-\tau} X^2(s)ds\int_0^{t-\tau}X^2(s+\tau)ds\] by the Cauchy-Schwartz inequality. Consequently, for \(t>\tau\) we obtain \[r_t(-\tau,0)^2\le \upsilon(t-\tau)(\upsilon(t)-\upsilon(\tau))<\upsilon(t-\tau)\upsilon(t),\] implying that \(\det (Q_t)> 0\).

Let \(p_\nu\) be the density of the measure \(\nu(t)\) for \(t>\tau\). In this example the evolution equation 35 reduces to \[\label{e:linear-evol} \frac{\partial}{\partial t} p(x,t)=-\frac{\partial}{\partial x}(axp(x,t))-\frac{\partial}{\partial x}\int_{\mathbb{R}} b y p_\nu(x,y,t)dy+\frac{1}{2}\frac{\partial^2}{\partial x^2}(\sigma^2p(x,t)).\tag{44}\] Since \(\nu(t)\) is a 2-dimensional Gaussian distribution, we can write \[\label{e:question} p_\nu(x,y,t)=p(x,t)p_\nu(y,t-\tau|x,t)\tag{45}\] where \(p_\nu(y,t-\tau|x,t)\), the density of \(x(t-\tau)\) conditional on \(x(t)=x\), is again Gaussian with mean \[m_{|}(x,t)=S_t\phi(-\tau)+\frac{r_t(-\tau,0)}{\upsilon(t)}[x-S_t\phi(0)].\] Consequently 44 takes the form \[\begin{align} \label{e:weakL1} \frac{\partial}{\partial t} p(x,t)=-\frac{\partial}{\partial x}([ax+b m_{|}(x,t)]p(x,t))+\frac{1}{2}\frac{\partial^2}{\partial x^2}(\sigma^2p(x,t)). \end{align}\tag{46}\] It follows that \(p\) satisfies Eq. 46 if and only if \[\label{e:sigr} \frac{d}{dt}\upsilon(t)=2a\upsilon(t)+2br_t(-\tau,0)+\sigma^2,\quad t>\tau.\tag{47}\] From 40 we see that 47 always holds.

Existence of stationary solutions for the linear first order example↩︎

Define \[\alpha_0=\max\{\mathrm{Re}\lambda: \lambda=a +be^{-\lambda \tau}\}.\] For each \(\alpha>\alpha_0\), there is a constant \(c\) such that \(|X(t)|\le c e^{\alpha t}\) for all \(t>0\) [37]. In particular, if \(\alpha_0<0\) then \(X(t)\) converges to zero exponentially rapidly.

Based on the work of [43], see also [37], we have \(\alpha_0<0\) if and only if \[\begin{align} \label{e:alpha0} a\tau <1,\quad b\tau + a\tau<0,\quad b\tau+a\tau \cos\kappa +\kappa\sin \kappa>0, \end{align}\tag{48}\] where \(\kappa\) is the root of \(\kappa=a\tau \tan \kappa\), \(0<\kappa<\pi\) if \(a\neq 0\) and \(\kappa=\pi/2\) if \(a=0\). These are values of \((a,b)\) inside the hatched wedge-shaped area of Figure 1.

In [45] it is shown that a stationary solution of 40 exists if and only if \(\alpha_0<0\) or, equivalently, the fundamental solution is square integrable: \[\label{e:ex} \int_0^\infty X^2(q)dq<\infty.\tag{49}\] Then we have \(\lim_{t\to\infty} S_t\phi(0)=0\) and \[\label{d:sigma} \lim_{t\to\infty} \upsilon(t)=\lim_{t\to\infty}\sigma^2 \int_0^{t} X^2(q)dq=\sigma^2\int_0^\infty X^2(q)dq=:\sigma_*^2\tag{50}\] leading to \[\lim_{t\to\infty} p(x,t)=f_*(x)\quad \text{for all }x\in \mathbb{R},\] where \[\label{eq:stat32den} f_*(x)=\frac{1}{\sqrt{2\pi\sigma_*^2}}e^{-\frac{x^2}{2\sigma_*^2}}.\tag{51}\]

If 49 holds, then according to [45] a stationary solution \(Z(t)\), \(t\ge -\tau\), of 40 can be represented as \[Z(t)=\sigma\int_{-\infty}^{t} X(t-s)dw(s),\quad t\in \mathbb{R},\] where \(X\) is the fundamental solution 39 , and \(w\) is extended to \(\mathbb{R}\). The covariance function of \(\{Z(t): t\in \mathbb{R}\}\) is given by \[K(t)=\mathbb{E}(Z(s)Z(t+s))=\sigma^2\int_{0}^\infty X(q)X(q+t)dq,\quad t\ge 0,\] with \(K(t):=K(-t)\) for \(t<0\). We see that \[K(0)=\sigma_*^2=\sigma^2 \int_0^\infty X^2(q)dq\quad \text{and}\quad K(\tau)=\lim_{t\to\infty}r_t(-\tau,0).\] Note that \[\label{e:ibv} 2a K(0)+2bK(\tau)+\sigma^2=0.\tag{52}\] To see this use the formula for \(K(\tau)\) and make use of 38 for the fundamental solution \(X\) to obtain \[bK(\tau)=\int_0^\infty b X(q-\tau)\sigma^2X(q)dq=\int_{0}^{\infty}(X'(q)-aX(q))\sigma^2X(q)dq,\] which gives \[bK(\tau)=\frac{\sigma^2}{2}\int_0^\infty \frac{d}{dq}[X^2(q)]dq -a K(0).\]

Observe that if \(\tau=0\) then condition 52 reduces to 22 . If \(\tau>0\) then the value of \(\sigma^2_*=K(0)\) can be calculated as a function of the parameters \(a,b,\sigma,\tau\) as in [45], where it is shown that \[K(t)=K(0)g_1(t)+K'(0)g_2(t),\quad t\in [0,\tau],\] with \[g_1(t)=\left\{ \begin{array}{ll} \cosh(lt), & if a^2>b^2, \\ 1, & if a^2=b^2, \\ \cos(lt), & if a^2<b^2, \end{array} \right. \quad g_2(t)=\left\{ \begin{array}{ll} \frac{1}{l}\sinh(lt), & if a^2>b^2, \\ t, & if a^2=b^2, \\ \frac{1}{l}\sin(lt), & if a^2<b^2, \end{array} \right.\] and \[l=\sqrt{|a^2-b^2|},\quad K(0)=\frac{\sigma^2}{2}\frac{bg_2(\tau)-1}{bg_1(\tau)+a},\quad K'(0)=-\frac{\sigma^2}{2}.\]

The Gibbs’ and conditional entropy in the linear first order case↩︎

Taking \(f_{m_1,\sigma_1^2}\) to be given by 42 and \(f_{m_2,\sigma_2^2}\) by 51 , we have \(\sigma_1^2 = \upsilon(t), m_1 = m_t(0)=S_t\phi(0), \sigma_2^2 = \sigma^2_*, m_2 = 0\) and thus it follows from 24 that \[H_G(p(\cdot,t|\phi)) = \dfrac 12 + \dfrac 12 \ln (2 \pi \upsilon(t))\] and, by 26 we have \[\label{dom32initial32condition} H_c(p(\cdot,t|\phi)|f_*) = \dfrac 12 \ln \dfrac{\upsilon(t)}{\sigma^2_*} + \dfrac 12 \left ( 1 - \dfrac{\upsilon(t)}{\sigma^2_*} \right) - \dfrac{[S_t\phi(0)]^2}{2 \sigma^2_*}.\tag{53}\] Consequently, it follows from 37 that for the density \(f\) of \(x(t)\) with initial condition with distribution \(\mu_0\) as defined in 36 we obtain the following lower bound \[H_c(f(\cdot,t)|f_*)\ge \frac{1}{2}\ln \frac{\upsilon(t)}{\sigma_*^2}+\frac{1}{2}\left(1-\frac{\upsilon(t)}{\sigma_*^2}\right)-\frac{1}{2\sigma_*^2}\int_C (S_t\phi(0))^2 \mu_0(d\phi).\label{e:ent-temp-inequald}\tag{54}\] Since \(S_t\phi(0)\to 0\) as \(t\to \infty\), we can conclude that \(H_c(f(t,\cdot)|f_*)\to 0\) for all initial distributions with finite second moment. Note that 54 gives an analogous lower bound for the entropy as in the non-delay case. We see that \(S_t\phi(0)\) and \(e^{ta}x_0\) are solutions of the corresponding deterministic differential equations, i.e. Eq. 40 and 19 with \(\sigma=0\).

If we calculate the temporal rate of change then for the Gibbs’ entropy \[\dfrac{dH_G(p(\cdot,t|\phi))}{dt} = \dfrac{1}{2 \upsilon(t)}\dfrac{d\upsilon(t)}{dt} \geq 0,\] since \(t\mapsto \upsilon(t)\) is an increasing and positive function of \(t\) by 41 . This is precisely the same conclusion reached in [2] for the Gibbs’ entropy rate of change when the initial variance is zero. For the conditional entropy \(H_c\) we obtain \[\dfrac{d H_c(p(\cdot,t|\phi)| f_*)}{dt}= \dfrac{1}{2 \upsilon(t)}\dfrac{d \upsilon(t)}{dt} \left(1-\frac{\upsilon(t)}{\sigma_*^2}\right) -\frac{1}{2\sigma_*^2}\frac{d}{dt}[S_t\phi(0)]^2. \label{e:cond32ent32derivative}\tag{55}\] Now note that since \(\upsilon(t) \leq \sigma^2_*\) by 50 , then for the first term in 55 \[\dfrac{1}{2 \upsilon(t)}\dfrac{d \upsilon(t)}{dt} \left(1-\frac{\upsilon(t)}{\sigma_*^2}\right) \geq 0.\] However we cannot assert that \[-\frac{1}{2\sigma_*^2}\frac{d}{dt}[S_t\phi(0)]^2 \geq 0,\] for the second term, and the problem is that \(S_t\phi(0)\) may exhibit a damped oscillation. For example, consider \(a=0\), \(b=-1\) and \(\tau=1.1\), and \(\phi(s)=1\). Then \(x'(t)=-1\) for \(t\in[0,\tau]\), so \(S_t\phi(0)=x(t)=1-t\) for \(t\in [0,\tau]\) and \[\frac{d}{dt}[S_t\phi(0)]^2=2x(t)x'(t)=-2(1-t)\] which is either positive or negative for \(t \in [0,1.1]\), see Figure 2.

Thus the existence of the delay in the dynamics has modified the temporal trajectory of the conditional entropy from being a monotone increase to equilibrium (in the zero delay case) to one in which the temporal behaviour may well be non-monotone in the approach to equilibrium and can be strongly dependent on the nature of the initial function \(\phi (s)\) for \(s \in [-\tau,0]\).

Lack of monotonicity of the entropies in one dimensional linear delay dynamics with and without noise↩︎

We now show that both the Gibbs’ entropy \(H_G(f(\cdot,t))\) and the conditional entropy \(H_c(f(\cdot,t)|f_*)\) may not be monotone functions of time. To see this we take as the initial distribution a Gaussian distribution. Suppose as in [1] that the initial condition \(x_0=\phi\) is a Gaussian process \(\xi\) with covariance function \(R_0\) of the form \[\label{e:R0} R_0(s_1,s_2)=\int_{-\tau}^{0}\eta_r(s_1)\eta_r(s_2)dr,\tag{56}\] for some function \(\eta_r\colon [-\tau,0]\to \mathbb{R}\) such that \((s,r)\mapsto \eta_r(s)\) is Borel measurable. Then the covariance \(R_t\) of the Gaussian process \(S_t\xi\) is given by \[R_t(s_1,s_2)=\int_{-\tau}^0 S_t\eta_r(s_1)S_t \eta_r(s_2)dr\] and in particular, for the variance \(\sigma^2(t)=R_t(0,0)\) of \(S_t\xi(0)\) we have \[\sigma^2(t)=\int_{-\tau}^0 (S_t\eta_r(0))^2dr.\]

One example of 56 is \(R_0(s_1,s_2)=\bar\sigma^2(\min\{s_1,s_2\}+\tau)\) for \(s_1,s_2\in [-\tau,0]\), where \(\bar\sigma>0\), since 56 holds with \[\eta_r(s)=\bar\sigma1_{[r,0]}(s),\quad s,r\in [-\tau,0].\] Then we have \(\xi(s)=\bar\sigma W(s+\tau)\), where \(W=\{W(t)\}_{t\ge 0}\) is a standard Wiener process on \([0,\infty)\). Consider now Eq. 40 with \(a=0\) \[\label{ex:lm} dx(t)=bx(t-\tau)dt+\sigma dw(t), \quad t>0,\tag{57}\] and the initial condition \[x(t)=\xi(t), \quad t\in [-\tau,0],\] where we assume that \(\xi\) is independent of the Wiener process \(w\). Then \[x(t)=S_t\xi(0)+y(t)\] has a Gaussian distribution with mean \(0\) and variance \(\bar\sigma^2_t\) being the sum of variances of \(S_t\xi(0)\) and \(y(t)\) \[\bar\sigma^2_t=\int_{-\tau}^0 (S_t\eta_r(0))^2dr+\upsilon(t).\] It is easily seen that \[S_t\eta_r(0)=\bar\sigma X(t-r), \quad t\ge 0, r\in[-\tau,0].\] Thus we obtain \[\label{sigD} \sigma^2(t)=\bar\sigma^2\int_{t}^{t+\tau} X^2(q)dq,\tag{58}\] leading to \[\bar\sigma^2_t=\frac{\bar\sigma^2}{\sigma^2}\upsilon(t+\tau)+\left(1-\frac{\bar\sigma^2}{\sigma^2}\right)\upsilon(t).\] Note that if \(\bar\sigma^2> \sigma^2\) then \(\bar\sigma^2_t\) need not to be a monotone function of time, implying that the Gibbs’s entropy and the conditional entropy of the density \(f(x,t)\) of the solution of 57 will not be monotone functions of time, see Figures 4 and 5.

Finally, let us recall that Eq. 57 has a stationary solution if and only if \[\label{climb} -\frac{\pi}{2}<b\tau<0.\tag{59}\] In that case we have \(f(x,t)\to f_*(x)\) as \(t\to \infty\) by 36 and the fact that \(p(x,t|\phi)\to f_*(x)\) as \(t\to \infty\) for each \(\phi\in C\) and arbitrary initial distribution.

Now, suppose that \(\sigma=0\) in 57 and that we start with an initial distribution being Gaussian as above. If condition 59 holds then the variance \(\sigma^2(t)\) of \(x(t)\) as given by Eq. 58 converges to \(0\) as \(t\to \infty\) and \(0\) is a stationary solution of 57 , see [1]. If \(b=-1\) and \(\tau=\pi/2\), then \(x(t)\) has a Gaussian density \(f(x,t)\) with variance \(\sigma^2(t)\). The graph of its Gibbs’ entropy is presented in Figure 6. It is known [1] that equation \(x'(t)=-x(t-\tau)\) with \(\tau=\pi/2\) has a non-zero stationary solution in which case \(f_*(x)\) is the Gaussian density with mean 0 and variance 1. Then the conditional entropy \(H_c(f(\cdot,t)|f_*)\) is of the form \[H_c(f(\cdot,t)|f_*)=\frac{1}{2}\ln \sigma^2(t)+\frac{1}{2}(1-\sigma^2(t))\] and its graph is presented in Figure 7.

7 Discussion↩︎

It is the universal experience that in all living things there is an inexorable one-way progression of events from birth through aging culminating in death. We never witness the reverse sequence. The rather astonishing thing, however, is that all of the laws (evolution equations) that are written down in physics show no preference for a direction of time. They are all equally valid for time \(t\) going in a positive direction as well as in a negative direction, and this is true for the equations of mechanics, electricity and magnetism, special and general relativity and quantum mechanics [46]. Why is there no clear preference for a direction of time in the dynamical equations of physics?

Countless scientists have thought about, and written about, this apparently contradictory situation. In a non-technical vein, [47] has written an extremely thoughtful essay examining a variety of issues related to this question that is informative, deep, and provocative. In his usual fashion, Martin [48] has an amusing survey of the problems that would be encountered if time could go backward that is well worth reading and pondering. Finally, [49] expounds very eloquently on the possibility that temporal irreversibility is cosmologically derived from the (observed) expansion of the universe. This is elaborated on in the essay by [50].

Many authors have considered the possible origins of the unidirectionality of time over the years, and without being exhaustive we mention [51], [52], [53], [46], [54] and [55] as those worth reading. There is, in addition, a very useful collection of reprinted musings on the subject in [56], as well as innumerable conference proceedings including [57], [58], [59], and [60].

Not surprisingly, since many of those who have considered the possible origins of temporal unidirectionality have been physicists, the issue of the Second Law of Thermodynamics has repeatedly been invoked, examined, and discussed. In the early part of the 20\(^{th}\) century [61] re-framed the issue of temporal unidirectionality in terms of the behaviour of entropy, saying:

“The law that entropy always increases holds, I think, the supreme position among the laws of Nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell’s equations - then so much the worse for Maxwell’s equations. If it is found to be contradicted by observation - well, these experimentalists do bungle things sometimes. But if your theory is found to be against the Second Law of Thermodynamics I can give you no hope; there is nothing for it to collapse in deepest humiliation."

Here we have explored only a few of the possibilities for the uni-directionality of time related to the temporal behaviour of entropy. Our considerations are not new except for the results of Section 6, but serve to highlight the nature of the problem. In Table 1 we have summarized our results here in terms of the temporal behaviours of \(H_G\), \(H_c\), and \(H_{NE}\).

The non-equilibrium Gibbs’ entropy \(H_G(f)\) is manifestly not a good candidate for \(S_{TD}(t)\) because its dynamical behavior is at odds with what is demanded by the Second Law of Thermodynamics. As we have demonstrated in [3] and here, concrete analytic examples can be constructed in which the direction of the temporal change in \(H_G(f)\) depends on the initial preparation of the system and others can be constructed in which \(H_G(f)\) oscillates in time. A number of other authors, including [62, pp. 122–129], Eq. 247, [35], and [16] have suggested that a time dependent entropy should be associated dynamically with \[\begin{align} H_{NE}(f) &\equiv& H_c(f|f_*e^{H_G(f_*)}) \nonumber \\ &=& H_c(f|f_*) + H_G(f_*) \end{align}\] as an extension of the [4] discussion of entropy. This also goes under the name of the “Gibbs’ entropy postulate" [63]–[68].

In the absence of both stochastic perturbation and delays, we have shown that the conditional entropy (and thus \(H_{NE}\)) is temporally constant. The introduction of stochasticity can induce the monotone approach of \(H_{NE}\) to a maximum of zero.

In our quest to extend the problem of entropy evolution to situations with delays, in Section 6 we have considered ‘density’ evolution in stochastically perturbed systems with delayed dynamics like 31 . We have derived a ‘Fokker-Planck’ like equation 35 for the ‘density’. This is exactly analogous to our procedure in [1] when we derived a ‘Liouville-like’ equation [1] for the density evolution under the action of completely deterministic delayed dynamics [1]. Both of these results for the Liouville and Fokker-Planck like evolution equations are equivalent to those derived by others, e.g. [41], [42] although our method of derivation deviates from theirs.⁵

The interesting finding is that the presence of delay can destroy this monotonicity and lead to an oscillatory behaviour of the Gibbs’ and conditional entropy, and thus \(H_{NE}\). If stochastic perturbations are simultaneously present then there may be an oscillatory approach of the entropies to a maximum. Thus it would seem that \(H_c\) and \(H_{NE}\) are not viable candidates for a time dependent non-equilibrium entropy if delays play a significant role in entropic behavior in the real world.

The next obvious extension will be to look at nonlinear delayed and stochastic delayed systems though it is not obvious to us how to proceed with this programme.

References↩︎