Dynamical phase transitions in certain non-ergodic stochastic processes

TL;DR

The paper identifies dynamical phase transitions in non-ergodic stochastic processes by studying singular large-deviation functions of time-integrated observables. It develops a unified framework based on backward Fokker-Planck and tilted-operator methods, showing that DPTs arise from competition between survival and diffusion, which manifests as crossings of leading eigenvalues in reducible tilted operators. The authors illustrate the mechanism with mortal Brownian motion and Brownian motion with absorbing boundaries, and extend the analysis to Markov chains with transient states, non-Markovian processes, and many-body contexts, supported by rare-event simulations. The work provides a general, robust picture for how dynamical phase structure emerges in a broad class of non-ergodic systems and suggests avenues for observing multiple DPTs in complex settings.

Abstract

We present a class of stochastic processes in which the large deviation functions of time-integrated observables exhibit singularities that relate to dynamical phase transitions of trajectories. These illustrative examples include Brownian motion with a death rate or in the presence of an absorbing wall, for which we consider a set of empirical observables such as the net displacement, local time, residence time, and area under the trajectory. Using a backward Fokker-Planck approach, we derive the large deviation functions of these observables, and demonstrate how singularities emerge from a competition between survival and diffusion. Furthermore, we analyse this scenario using an alternative approach with tilted operators, showing that at the singular point, the effective dynamics undergoes an abrupt transition. Extending this approach, we show that similar transitions may generically arise in Markov chains with transient states. This scenario is robust and generalizable for non-Markovian dynamics and for many-body systems, potentially leading to multiple dynamical phase transitions. We have confirmed most of our findings on the singular large-deviation function using rare-event simulation techniques.

Figures (23)

• Figure 1: (color online) The solid line denotes the ldf in \ref{['eq:phi sticky 1']} for the distribution of position of a mortal Brownian particle with a death rate $\alpha=1$. The change of color at $q=\pm q_c$ with $q_c=\sqrt{4\alpha}$ highlights the second-derivative-discontinuity of $\phi(q)$ in \ref{['eq:phi sticky 1']}. The dots on the plot represent results obtained by importance sampling simulations for $T=100$ and $dt=0.01$ (see Appendix \ref{['app:importance sampling']}).
• Figure 2: (color online) The ldf in \ref{['eq:phi sticky area']} for the area under alive trajectories of a mortal Brownian particle with a death rate $\alpha=1$. The change of color at $q=\pm q_c$, where $q_c=\frac{2}{3}\sqrt{\alpha}$, signifies the second-derivative-discontinuity of $\phi(q)$ in \ref{['eq:phi sticky 1']}. The data points indicate results from importance sampling simulations for $T = 100$ with $dt = 0.01$ (see Appendix. \ref{['app:importance sampling']}). They slightly deviate from the result of \ref{['eq:phi sticky area']} because the asymptotic limit $T\to\infty$ has not been reached yet, see the dashed green line corresponding to the exact ldf computed from \ref{['eqn: stickydistribution_area']}.
• Figure 3: (color online) The blue–pink curve shows the ldf in \ref{['eq:ldf local mortal']} for the local time at the origin of a mortal Brownian particle with death rate $\alpha=1$, starting at the origin. The linear part (colored blue) meets the non-linear part (colored pink) at a singular point $q_c=1$. Uniformly colored curves correspond to ldfs extracted from the exact distribution $P_T(Q\vert 0)$ expressed in terms of the scaling function $h(x,t)$ in \ref{['eq:exact h']}, for $T=1$ (lightest), $T=2$ (intermediate), and $T=10$ (darkest). The inset shows the associated scgf from from \ref{['eq:scgf local time mortal BM']}, with singularity at $p_c=2$.
• Figure 4: (color online) The solid line represents the scgf in \ref{['eq:cgf sticky residence']} for the residence time in the interval $[-1,1]$ of a mortal Brownian particle with a death rate $\alpha=1$. The scgf is the maximum of the two poles: $\mu_1=0$ (blue) and $\mu_2(p)=s^\star(p)$ (pink) of $R_s$ in \ref{['eq:R pole residence sticky']}, leading to a singularity at $p_c=1.74$. The pink curve meets y-axis at value $\mu_2(0)=-\alpha$ which can be seen from the largest root $s^\star$ of $d(s,0)$ in \ref{['eq:dsp']}.
• Figure 5: (color online) The ldf for residence time of a mortal Brownian particle with the same parameter values of Fig. \ref{['fig:sticky_residence_scgf']}. The solid line represents the theoretical result in \ref{['eq:ldf sticky residence']} while the points indicate results from importance sampling simulations with parameter values $T = 100$, $dt = 0.01$ (see Appendix. \ref{['app:importance sampling']}). At the singular point $q_c=0.78$, marked by the dashed line, the $\phi"(q)$ is discontinuous.