Gallavotti's equivalence conjecture for reversible and irreversible thermostatted ensembles

Let $N$ particles occupy a volume $V$, with density $d=N/V$. Let $\m_{\d,T}$ be the SRB distribution of the reversible Gaussian-thermostatted model at kinetic-energy parameter $T$, and let $\widetilde \m_{\d, }$ be the SRB distribution of the irreversible friction model at density $\d$ and friction coefficient $$. The parameters correspond when

$$\d=\d',\qquad \frac32 k_B T=\int \frac12\left(\sum_j m\dot{\V x}_j^2\right)\,\widetilde \m_{\d, }(d\V x\,d\dot{\V x}).$$

A local observable is an observable depending only on the microscopic state of electrons whose positions lie inside a fixed box $V_0$, and let ${\cal L}$ denote the local smooth observables. Equivalence conjecture. For every $F\in{\cal L}$, if $T$ and $$ are related by the correspondence above, then

$$\lim_{N\to\infty,\,N/V=\d}\widetilde \m_{\d, }(F)=\lim_{N\to\infty,\,N/V=\d}\m_{\d,T}(F).$$

The conjecture proposes equivalence of the reversible and irreversible nonequilibrium statistical ensembles for local observables, analogous to canonical–microcanonical equivalence in equilibrium statistical mechanics. The source reports arguments in its favor and relates it to ergodicity and spatial translation invariance, but gives no resolution.