Non-equivalence of Dynamical Ensembles and Emergent Non-ergodicity

Abstract

Dynamical ensembles have been introduced to study constrained stochastic processes. In the microcanonical ensemble, the value of a dynamical observable is constrained to a given value. In the canonical ensemble a bias is introduced in the process to move the mean value of this observable. The equivalence between the two ensembles means that calculations in one or the other ensemble lead to the same result. In this paper, we study the physical conditions associated with ensemble equivalence and the consequences of non-equivalence. For continuous time Markov jump processes, we show that ergodicity guarantees ensemble equivalence. For non-ergodic systems or systems with emergent ergodicity breaking, we adapt a method developed for equilibrium ensembles to compute asymptotic probabilities while caring about the initial condition. We illustrate our results on the infinite range Ising model by characterizing the fluctuations of magnetization and activity. We discuss the emergence of non-ergodicity by showing that the initial condition can only be forgotten after a time that scales exponentially with the number of spins.

Appendices

Appendix A: Derivation of the Propagator for Generating Function

We look for an expression of the propagator of the generating function $G(x_{\mathrm {f}},x_{\mathrm {i}},\kappa ,\gamma )= \big \langle e^{NT(\kappa {m}+\gamma a)}\big \rangle _{x_{\mathrm {f}},x_{\mathrm {i}}}$ as a path integral over all trajectories for the system state n. We use the discrete time $t_k=k \mathrm {d}t$ corresponding to the k’s time step of duration $\mathrm {d}t$, with the final time $T=t_L = L \mathrm {d}t$. We write the system state $n_{t_k}=n_k$ for short. This state at time $t_k$ changes by $n_{k}-n_{k-1}=2\epsilon _k = -2, 0$ or 2 when a spin jumps from up to down, doesn’t jump or jumps from down to up. The sum over all paths for a given initial condition is

$$\begin{aligned} \sum _{[\epsilon _k]} = \sum _{\epsilon _0}\sum _{\epsilon _1} \cdots \sum _{\epsilon _{N-1}}. \end{aligned}$$

(85)

Since jump probabilities depend on the magnetization, the path integration should include a sum on the system state $n_k$ for all k

$$\begin{aligned} \sum _{[n_k]}= \sum _{n_1}\sum _{n_2} \cdots \sum _{n_{L-1}}, \end{aligned}$$

(86)

with the condition that one spin at most jumps at each time step. In the above sums, the index n goes from $-N$ to N. All in all, the propagator of the generating function writes

$$\begin{aligned} G(n_{\mathrm {i}},n_{\mathrm {f}},\kappa ,\gamma )= & {} \sum _{[n_k,\epsilon _k]} \left[ \prod _{k=1}^{L} p(n_{k}|n_{k-1}) \delta (n_{k}-n_{k-1}-2\epsilon _{k-1}) \right] \nonumber \\&\times \,e^{\kappa \left( \sum _{k=1}^L n_{k-1} \mathrm {d}t \right) +\gamma \left( \sum _{k=1\, ,\epsilon _k \ne 0}^L 1 \right) }, \end{aligned}$$

(87)

with $p(n_{k}|n_{k-1})$ the probability of the transition $n_{k-1} \rightarrow n_{k}$. Let’s use the Laplace representation of the Dirac distribution

$$\begin{aligned} \delta (n_{k}-n_{k-1}-2\epsilon _{k-1}) = \int _{i{b}_k-\infty }^{i{b}_k+\infty } \frac{\mathrm {d}\mathfrak {p}_{k}}{2 \pi }e^{ i \mathfrak {p}_{k}(n_{k}-n_{k-1}-2\epsilon _{k-1})}, \quad {b}_k \in \mathbb {R}, \end{aligned}$$

(88)

for every integer k between 1 and L to get

$$\begin{aligned} G(n_{\mathrm {i}},n_{\mathrm {f}},\kappa ,\gamma )= & {} \sum _{[n_k,\epsilon _k]} \int \left( \prod _{k=1}^{L} \frac{\mathrm {d}\mathfrak {p}_k}{2 \pi } \right) \nonumber \\&\times \,\left[ \prod _{k=1}^{L} p(n_{k}|n_{k-1}) e^{i \mathfrak {p}_k(n_{k}-n_{k-1}-2\epsilon _{k-1}) +\kappa n_{k-1} \mathrm {d}t + \gamma } \right] . \end{aligned}$$

(89)

In the next step, we sum over $[\epsilon _k]$ to obtain

$$\begin{aligned} G(n_{\mathrm {i}},n_{\mathrm {f}},\kappa ,\gamma )= & {} \sum _{[n_k]} \int \left( \prod _{k=1}^{L} \frac{\mathrm {d}\mathfrak {p}_k}{2 \pi } e^{i \mathfrak {p}_k(n_{k}-n_{k-1})} \right) \nonumber \\&\times \,\prod _{k=1}^{L} \left[ p(n_{k-1}|n_{k-1}) + \sum _{\epsilon =\pm 1} p(n_{k-1}+\epsilon |n_{k-1}) e^{-2i\mathfrak {p}_k \epsilon +\gamma } \right] e^{\kappa n_{k-1}\mathrm {d}t}.\nonumber \\ \end{aligned}$$

(90)

The propagator for staying in the same state during a time step is $p(n_{k}|n_{k}) = 1- \left( \sum \limits _{\epsilon =\pm 1}K(n_k+2\epsilon ,n_k)\right) \mathrm {d}t$, and when changing of state writes $p(n_{k}+\epsilon |n_{k}) = K(n_k+2\epsilon ,n_k) \mathrm {d}t$. Exponentiating the product yields

$$\begin{aligned} G(n_{\mathrm {i}},n_{\mathrm {f}},\kappa ,\gamma )= & {} \sum _{[n_k]} \int \left( \prod _{k=1}^{L} \frac{\mathrm {d}\mathfrak {p}_k}{2 \pi } \right) \exp { \left( i \mathrm {d}t \sum _{k=1}^{L} \mathfrak {p}_k \mathchoice{\dfrac{(n_{k}-n_{k-1})}{\mathrm {d}t}}{\dfrac{(n_{k}-n_{k-1})}{\mathrm {d}t}}{\frac{(n_{k}-n_{k-1})}{\mathrm {d}t}}{\frac{(n_{k}-n_{k-1})}{\mathrm {d}t}} \right) } \nonumber \\&\times \,\exp \left[ \sum _{k=1}^{L} \mathrm {d}t \left( \sum _{\epsilon } K(n_{k-1}+2\epsilon ,n_{k-1}) \left( e^{-2i \mathfrak {p}_k\epsilon + \gamma } -1 \right) +\kappa n_{k-1} \right) \right] .\nonumber \\ \end{aligned}$$

(91)

The propagator of the generating function becomes in the continuous limit

$$\begin{aligned} G(x_{\mathrm {f}},x_{\mathrm {i}},\kappa ,\gamma )= & {} \int \mathcal {D} [x] \int \mathcal {D}[\mathfrak {p}] \exp \left[ N \int _{0}^{T} \mathrm {d}t i \mathfrak {p}_t \dot{x}_{t} \right. \nonumber \\&\left. - \int _{0}^{T} \mathrm {d}t \left( \sum _{\epsilon } J_\epsilon (x_{t}) \left( 1- e^{-2 i \mathfrak {p}_t\epsilon + \gamma } \right) +\kappa x_{t} \right) \right] \end{aligned}$$

(92)

where $\int \mathcal {D} [x]$ (resp. $\int \mathcal {D} [\mathfrak {p}]$) is a short notation for the integral on the paths of density (resp. momentum) with given initial and final values. Finally, the path integral expression of the generating function is given by

$$\begin{aligned} G(x_{\mathrm {f}},x_{\mathrm {i}},\kappa ,\gamma )= \int \mathcal {D} [x] \int \mathcal {D}[\mathfrak {p}] \exp \left( N \mathcal {A}_{\kappa ,\gamma } [x,\mathfrak {p},T] \right) \end{aligned}$$

(93)

with the action $\mathcal {A}_{\kappa ,\gamma }$ being a functional of the paths $[x,\mathfrak {p}]$

$$\begin{aligned} \mathcal {A}_{\kappa ,\gamma } [x,\mathfrak {p},T] = \int _{0}^{T} \mathrm {d}t i \mathfrak {p}_t \dot{x}_{t} - \int _{0}^{T} \mathrm {d}t \mathcal {H}(x_{t},\mathfrak {p}_t,\kappa ,\gamma ). \end{aligned}$$

(94)

We have introduced the Hamiltonian function

$$\begin{aligned} \mathcal {H}(x,\mathfrak {p},\kappa ,\gamma ) = \sum _{\epsilon =\pm 1} \left[ 1-\exp { \left( -2 i \mathfrak {p} \epsilon +\gamma \right) } \right] J_\epsilon (x) -\kappa x. \end{aligned}$$

(95)

This function is the representation in the continuous limit of a tridiagonal Metzler matrix, whose spectrum is real.² As a consequence, $i \mathfrak {p}$ is a real number.

Appendix B: Wentzel–Kramers–Brillouin Approximation

We now use a saddle point integration to obtain the leading order in N of the propagator of the generating function $G(x_{\mathrm {f}},x_{\mathrm {i}},\kappa ,\gamma )$ which is asymptotically exact in the large size limit. For each integration, we deform the contour so that it goes through the saddle point of $\mathcal {A}_{\kappa ,\gamma } [x,\mathfrak {p},t]$. Therefore, the saddle point is the minimum of $\mathcal {A}_{\kappa ,\gamma } [x,\mathfrak {p},t]$ solving

$$\begin{aligned} \mathchoice{\dfrac{\delta }{\delta \mathfrak {p}_t}}{\dfrac{\delta }{\delta \mathfrak {p}_t}}{\frac{\delta }{\delta \mathfrak {p}_t}}{\frac{\delta }{\delta \mathfrak {p}_t}} \mathcal {A}_{\kappa ,\gamma } [x,\mathfrak {p},T] = 0, \end{aligned}$$

(96)

$$\begin{aligned} \mathchoice{\dfrac{\delta }{\delta x_t}}{\dfrac{\delta }{\delta x_t}}{\frac{\delta }{\delta x_t}}{\frac{\delta }{\delta x_t}} \mathcal {A}_{\kappa ,\gamma } [x,\mathfrak {p},T] = 0, \end{aligned}$$

(97)

where $\mathchoice{\dfrac{\delta }{\delta x_t}}{\dfrac{\delta }{\delta x_t}}{\frac{\delta }{\delta x_t}}{\frac{\delta }{\delta x_t}}$ denotes a functional derivative.

In this case, the saddle point calculation is equivalent to the WKB approximation of quantum mechanics. The propagator of the generating function will be given by the paths maximizing the action, i.e. the classical paths starting at $x_{\mathrm {i}}$ and ending at $x_{\mathrm {f}}$. These paths solve the Hamilton’s equations

$$\begin{aligned} i \dot{x}_t =&\,\partial _\mathfrak {p} \mathcal {H}(x_t,\mathfrak {p}_t,\kappa ,\gamma ), \end{aligned}$$

(98)

$$\begin{aligned} i \dot{\mathfrak {p}}_t =&- \partial _x \mathcal {H}(x_t,\mathfrak {p}_t,\kappa ,\gamma ). \end{aligned}$$

(99)

The propagator of the generating function can be written as

$$\begin{aligned} G(x_{\mathrm {f}},x_{\mathrm {i}},\kappa ,\gamma ) \simeq _{ N \rightarrow +\infty } \exp \left[ N\int _{0}^{T} \mathrm {d}t \left( i \mathfrak {p}_t \dot{x}_{t} - \mathcal {H}(x_{t},\mathfrak {p}_t,\kappa ,\gamma ) \right) \right] , \end{aligned}$$

(100)

where $(x,\mathfrak {p})$ correspond to the solutions of Eqs. (98)–(99) with initial and final conditions $x_{\mathrm {i}}$ and $x_{\mathrm {f}}$. Moreover the Hamiltonian (95) is time independent, and is hence a conserved quantity along the trajectory. Maupertuis’ action $\int _{0}^{T} \mathrm {d}t i \mathfrak {p}_t \dot{x}_{t}$ is a function of the Hamiltonian and writes

$$\begin{aligned} \int _{0}^{T}\mathrm {d}t i \mathfrak {p}_t \dot{x}_{t} = \int _{x_{\mathrm {i}}}^{x_{\mathrm {f}}}\mathrm {d}x i \mathfrak {p}(x). \end{aligned}$$

(101)

where $\mathfrak {p}(x)$ can be obtained by inverting Eq. (95). We observe that Maupertuis’ action is always negative in view of the clockwise motion along the orbits that can be justified by inspection of Eq. (98).

In the large size limit, we can also use a saddle point integration on $\mathfrak {p}$ to switch to the Lagrangian framework. To do so, we look for the solution of Eq. (98). We derive a quadratic equation either for $e^{i\mathfrak {p}_{s}}$ or for $e^{-i\mathfrak {p}_{s}}$ whose solutions are

$$\begin{aligned} e^{\pm 2i\mathfrak {p}_{s}}= \mathchoice{\dfrac{\mp \dot{x}_{s} + \sqrt{\dot{x_{s}}^2+\varphi (x_{s},\gamma )}}{2 J_\mp (x_{s}) e^{\gamma }}}{\dfrac{\mp \dot{x}_{s} + \sqrt{\dot{x_{s}}^2+\varphi (x_{s},\gamma )}}{2 J_\mp (x_{s}) e^{\gamma }}}{\frac{\mp \dot{x}_{s} + \sqrt{\dot{x_{s}}^2+\varphi (x_{s},\gamma )}}{2 J_\mp (x_{s}) e^{\gamma }}}{\frac{\mp \dot{x}_{s} + \sqrt{\dot{x_{s}}^2+\varphi (x_{s},\gamma )}}{2 J_\mp (x_{s}) e^{\gamma }}} , \end{aligned}$$

(102)

with $\varphi (x,\gamma )$ defined in Eq. (59). Inserting this into Eq. (93) leads to the propagator of the generating function of Eq. (57) with Lagrangian (58).