Entropy production in systems with random transition rates close to equilibrium

We study the entropy production of systems out of equilibrium using random networks. We focus on systems with a finite number of states described by a master equation close to equilibrium. The dynamics are mapped into a network of states (nodes) connected by transition rates (links). Using this framework, we analyze the entropy production of ensembles of randomly generated networks owing to specific constraints (e.g., size or symmetries) and identify the most important parameters that determine its value. This analysis gives a null-model estimation for the entropy production that can be used for comparison with specific systems.

Figure 1

(Left) A master equation (ME) system can be represented by a graph in which nodes correspond to states and links to transition rates between them (sketched here as a six-node network). An external probability current $J$ enters into the system by one of the states and exits from another one to ensure stationarity. We analyze the total entropy production when the system is close to equilibrium. (Right) In symmetric ME systems ($w_{i\to j}=w_{j\to i}$), the entropy production can be computed from an equivalent network composed exclusively by the external nodes (the ones coupled to the environment via the external current) linked by a symmetric transition rate of strength $w_{\mathrm{eq}}$. Equivalently, the entropy production of slightly asymmetric networks can be found from an equivalent system composed by the external nodes and two asymmetric links $w_{\mathrm{eq}}(1\pm \sigma {\epsilon }_{\mathrm{eq}})$. Both $w_{\mathrm{eq}}$ and ${\epsilon }_{\mathrm{eq}}$ encode the topological structure of the underlying network.

Figure 2

Entropy production of ME systems represented by random (Erdős-Renyi) networks, compared to the theoretical prediction given by Eq. (7). Transition rates $w_{i\to j}$ are randomly generated from a Gaussian distribution with mean $w=1$ and standard deviation $w\sigma$. Each data set is composed by 100 realizations of networks of size $N=25$, fixing the parameter ${\omega }_{1\to N}=10$. The external flux $J$ is a random variable, being ${log}_{10}J$ uniformly distributed in $[-4,-1]$ (so that $J\in [{10}^{-4},{10}^{-1}]$). All plots are in log-log scale. (Top panel) Symmetric networks, for which $w_{i\to j}=w_{j\to i}$ for each pair of links. The connectivity $K$ is a uniform random variable in $K\in [0.25,1]$. (Bottom panel) Nonsymmetric networks: we remove the symmetric constraint, and therefore $w_{i\to j}\ne w_{j\to i}$ in general. In this case, we have fixed the connectivity $K=0.5$. In both panels, insets represent the same corresponding points but using a color scale for the external probability flux $J$. Similar results can be obtained for other network topologies different from the Erdős-Renyi [Eq. (7) is valid in general].

Figure 3

Deviations from Eq. (7) follow generic scaling relations. The relation between ${\dot{S}}^{*}$ and $J$ for networks generated with different $N, K, J$, and $\sigma$ can be enlightened plotting ${\dot{S}}^{*}-\langle \delta {\dot{S}}_{\mathcal{J}}\rangle$ against $\langle {\dot{S}}_{\mathcal{J}}\rangle$. All data sets collapse into the straight line $y=x$ (with deviations around). We have generated 250 networks for each system size with a random connectivity uniformly distributed in $K\in [0.25,1]$ and a external flux $J$ such that ${log}_{10}J$ is a uniform random variable in $[-4,-1]$ ($J\in [{10}^{-4},{10}^{-1}]$); the heterogeneity has been taken $\sigma =J$ for each realization (this ensures that the current and heterogeneity contributions to the entropy production are of the same order and therefore both have to be considered). (Inset) We collapse the PDFs of $\delta {\dot{S}}_{\mathcal{J}}$ by subtracting the contribution given by Joule's law, Eq. (7), and the mean $w[KN-(2+K)]{\sigma }^2$, and dividing by the standard deviation $w\sqrt{4K}{\sigma }^2$ (blue, green, yellow, and red points correspond to $K=0.25,0.5,0.75$, and 1, respectively). The corresponding $w_{\mathrm{eq}}$ is computed setting $w_{i\to j}=w$ for each network topology. All histograms collapse into a Gaussian distribution of zero mean and unit variance (dashed line). We have generated ${10}^4$ independent realizations for each set of parameters, setting $J=\sigma ={10}^{-3}$. We have set ${\omega }_{1\to N}=10$ in both panels.

Figure 4

We generate random Erdős-Renyi topologies of mean connectivity $K$ and size $N$, where each link $w_{i\to j}$ is an independent Gaussian variable of mean $w$ and standard deviation $\sigma w$. We set $w=1$ and $\sigma ={10}^{-2}$. In the absence of external flux ($J=0$), we integrate numerically the associated ME until stationarity and calculate $q_i=(N{p}_i^{*}-1)/\sigma$. (Top panel) We compare the numerical value of $q_i$ with the one given by the approximation of Eq. (B10), for all the nodes in the network and for ${10}^3$ randomly generated networks. Network size has been set to $N=100$. (Bottom panel) PDF of the error made in Eq. (B10), rescaled with the system size $N$, for $N=50$ (circles), $N=100$ (solid line), and $N=200$ (triangles), for different connectivities with the same color code as on the top panel. For each connectivity, the PDFs of the rescaled error $N\mathrm{\Delta }{q}_i$ collapse into a single curve, illustrating that the error $\mathrm{\Delta }{q}_i=O\left(N^{-1}\right)$.

Figure 5

We generate random Erdős-Renyi topologies of mean connectivity $K$ and size $N$, where each link $w_{i\to j}=w=1$ (i.e., no asymmetry). The external flux is set to $J={10}^{-2}$. We integrate numerically the associated ME until stationarity and calculate $r_i=(N{p}_i^{*}-1)/J$. (Top panel) We compare the numerical value of $r_i$ with the one given by the approximation of Eq. (B15), for all the nodes in the network and for ${10}^3$ randomly generated networks. Network size has been set to $N=100$. (Bottom panel) PDF of the error made in Eq. (B15), rescaled with the system size, for $N=50$ (circles), $N=100$ (solid line), and $N=200$ (triangles), for different connectivities with the same color code as on the top panel. In this case, we were able to capture also the scaling with the connectivity $K$. When rescaled properly, the location of the peaks approximately collapses, illustrating that the error made is $\mathrm{\Delta }{r}_i=O\left(K^{-2}{N}^{-1}\right)$.

Figure 6

Comparison between $w_{\mathrm{eq}}$ with the approximation given by Eq. (C1) in symmetric ME systems with networks of size $N=50,100,200$ and connectivity $K=0.25,0.5,0.75,1$. All existing links in the network has a rate value $w=1$. For each network, $w_{\mathrm{eq}}$ has been obtained by integrating numerically the ME with an additional external flux $J=0.01$, and then calculating $w_{\mathrm{eq}}=(p_1^{*}-p_N^{*})/J$.

Figure 7

Collapse of the PDF for 9 data sets of $w_{\mathrm{eq}}$, for $K=0.25$ (blue), $K=0.5$ (green), and $K=0.75$ (yellow), and for each one, for $N=50$ (circles), $N=100$ (triangles), and $N=200$ (squares) (same data sets of Fig. 6), obtained by subtracting the corresponding mean $NKw/2$ and dividing by the standard deviation $NK(1-K)w/8$ on each data [Eqs. (C8) and (C9) for the top panel and Eqs. (C10) and (C11) for the bottom one]. Dashed lines represent a Gaussian distribution of zero mean and unit variance. Deviations stem from finite size corrections, and indeed PDFs for larger system sizes fit better the Gaussian distribution.

Figure 8

Collapse of different $P\left({\epsilon }_{\mathrm{eq}}\right)$ for asymmetric ME systems in ensembles of ${10}^4$ networks of size $N=50$ (circles), $N=100$ (triangles), and $N=200$ (squares), and for each one, for connectivities $K=0.25$ (blue), $K=0.5$ (green), and $K=0.75$ (yellow). Link weights in the network are independent Gaussian variables of mean $w=1$ and standard deviation $w\sigma =0.01$. For each network, ${\epsilon }_{\mathrm{eq}}$ has been obtained by integrating numerically the ME with the external flux $J=0$, and then calculating ${\epsilon }_{\mathrm{eq}}=(p_1^{*}-p_N^{*})/\sigma$. Dark dashed line represents the Gaussian distribution of zero mean and unit variance. The collapse has been obtained by dividing each value of ${\epsilon }_{\mathrm{eq}}$ by its corresponding standard deviation obtained analytically, $2/\left(N\sqrt{KN}\right)$.