A Low Temperature Analysis of the Boundary Driven Kawasaki Process

Abstract

Low temperature analysis of nonequilibrium systems requires finding the states with the longest lifetime and that are most accessible from other states. We determine these dominant states for a one-dimensional diffusive lattice gas subject to exclusion and with nearest neighbor interaction. They do not correspond to lowest energy configurations even though the particle current tends to zero as the temperature reaches zero. That is because the dynamical activity that sets the effective time scale, also goes to zero with temperature. The result is a non-trivial asymptotic phase diagram, which crucially depends on the interaction coupling and the relative chemical potentials of the reservoirs.

Appendices

Appendix A: Kirchhoff Formula

A useful representation of the stationary distribution is in terms of the Kirchhoff formula [2, 17, 19] and [23]

$$ \rho(x)=\frac{W(x)}{\sum_y W(y)}, \quad W(x)=\sum_{\mathcal{T}}w(\mathcal{T}_x) $$

(25)

in which the last sum runs over all spanning trees in the graph $\mathcal{G}$ and $\mathcal{T}_{x}$ denotes the in-tree to x defined for any tree $\mathcal{T}$ and state x by orienting every edge in $\mathcal{T}$ towards x; its weight $w(\mathcal{T}_{x})$ is

$$ w(\mathcal{T}_x):=\prod_{b\in\mathcal{T}_x} k(b) $$

(26)

i.e., the product of transition rates k(b)=k(y,z) over all oriented edges b≡(y,z) in the in-tree $\mathcal{T}_{x}$. (To simplify notation, we identify the graph (see Fig. 4) with the set of its edges. We refer to [2] for the necessary elements of graph theory.)

Fig. 4

A graph consisting of 4 states where the collection of red (thick) arrows is one of the possible in-trees $\mathcal{T}_{x}$. The collection of black (dashed) arrows indicate other possible transitions on the graph G

Appendix B: Matrix Calculation

One can easily check from DE=D+E−ED that for all n≥0

$$ DE^{n}=(E+1)^nD+E(E+1)^n-E^{n+1} = D + \sum_{i,k} c_{ik} E^i D^k $$

(27)

for positive coefficients c _ik . Similarly for m≥0

$$ D^mE=E(1+D)^m+D(1+D)^m-D^{m+1} = E + \sum_{i,k} f_{ik} E^i D^k $$

(28)

for positive coefficients f _ik . Therefore,

$$\begin{aligned} \langle W\vert D W_{D,E} E\vert V \rangle =&\sum _{k,l}c_{k,l} \langle W\vert D E^k D^l E\vert V\rangle \\ =&\sum_{k,l}c_{k,l} \langle W \vert\bigl( (E+1)^kD+E(E+1)^k-E^{k+1} \bigr) \\ & {}\times \bigl(E(1+D)^l+D(1+D)^l-D^{l+1} \bigr)\vert V\rangle \\ \cong& \biggl(c_{0,0} \biggl(\frac{1}{\alpha(\beta)}+\frac{1}{\sigma(\beta)} \biggr)+O\bigl(e^{-2\beta}\bigr) \biggr)\langle W\vert V\rangle \end{aligned}$$

(29)

where the third equality follows from retaining only the lowest order terms in (27), (28).