Intensive thermodynamic parameters in nonequilibrium systems

Considering a broad class of steady-state nonequilibrium systems for which some additive quantities are conserved by the dynamics, we introduce from a statistical approach intensive thermodynamic parameters (ITPs) conjugated to the conserved quantities. This definition does not require any detailed balance relation to be fulfilled. Rather, the system must satisfy a general additivity property, which holds in most of the models usually considered in the literature, including those described by a matrix product ansatz with finite matrices. The main property of these ITPs is to take equal values in two subsystems, making them a powerful tool to describe nonequilibrium phase coexistence, as illustrated on different models. We finally discuss the issue of the equalization of ITPs when two different systems are put into contact. This issue is closely related to the possibility of measuring the ITPs using a small auxiliary system, in the same way as temperature is measured with a thermometer, and points at one of the major difficulties of nonequilibrium statistical mechanics. In addition, an efficient alternative determination, based on the measure of fluctuations, is also proposed and illustrated.

Figure 1

Illustration of two mass transport models with periodic boundary conditions, connected through a contact made of a single additional link. A zoom over the contact is also shown in the dashed frame; arrows indicate the oriented links through which mass is transported.

Figure 2

Time dependence of the masses of the condensates (a), the ITPs (b) and the densities of the fluid phases (c) in two mass transport models in contact, ${\mathcal{S}}_1$ (solid line) and ${\mathcal{S}}_2$ (dashed line)—see Fig. 1. Both of them contain an impurity site. Parameter values are ${\xi }_1=1.5$ and ${\xi }_2=1$ for the impurities, ${\eta }_1=2$ and ${\eta }_2=1$ for the homogeneous parts—see text. The systems are separately in steady state (with overall initial densities ${\rho }_1={\rho }_2=2$) before the contact is established at time $t=0$ (dotted line). Note that the ITPs eventually equalize, and that the difference between the fluid densities increases after the contact is established.

Figure 3

Schematic drawing of two generic systems in contact. Arrows illustrate the possibility to transfer a conserved quantity between the two systems. Contrary to Fig. 1, the contact generally extends over many sites.

Figure 4

Measurement process on a system ${\mathcal{S}}_1$ using an ITP meter ${\mathcal{S}}_2$. The values of the ITPs ${\lambda }_1$ (dashed line) and ${\lambda }_2$ (solid line) are plotted versus time. The systems are characterized by parameters ${\eta }_1=3$, ${\eta }_2=5$ (see text). The contact is switched on at time $t=0$. Initially, ${\rho }_1={\rho }_2=10$, so that ${\lambda }_1=0.3$ and ${\lambda }_2=0.5$. Note that ${\lambda }_1$ almost stays constant whereas after a sufficient time ${\lambda }_2$ converges to ${\lambda }_1$.

Figure 5

Equation of state $\lambda$ as a function of $\rho$ in two mass transport models, obtained by numerical simulations using the measurement procedure described in the text (×), compared with the theoretical prediction $\rho \left(\lambda \right)$ derived in the grand-canonical ensemble (solid line); (a) fully factorized steady state with $f\left(m\right)={(1+m)}^2$, (b) pair-factorized case with $g(m,n)=m+n$. Insets: dependence of the variance $V\left(M_N\right)$ on the subsystem size $N$ (∗) for densities $\rho =1.2$ (a) and $\rho =0.5$ (b); solid lines are linear fits.