Entropy Production in Exactly Solvable Systems

Abstract

The rate of entropy production by a stochastic process quantifies how far it is from thermodynamic equilibrium. Equivalently, entropy production captures the degree to which global detailed balance and time-reversal symmetry are broken. Despite abundant references to entropy production in the literature and its many applications in the study of non-equilibrium stochastic particle systems, a comprehensive list of typical examples illustrating the fundamentals of entropy production is lacking. Here, we present a brief, self-contained review of entropy production and calculate it from first principles in a catalogue of exactly solvable setups, encompassing both discrete- and continuous-state Markov processes, as well as single- and multiple-particle systems. The examples covered in this work provide a stepping stone for further studies on entropy production of more complex systems, such as many-particle active matter, as well as a benchmark for the development of alternative mathematical formalisms.

Run-and-tumble motion in a harmonic potential: field theory and entropy production

“…If there is no tumbling (α = 0), then the system is an Ornstein-Uhlenbeck process, which is at equilibrium and therefore produces no entropy. The positive entropy production rate implies the breaking of time symmetry whereby forward and backward trajectories are distinguishable [46]. This is visible in the trajectory of an RnT particle, such as in figure 1, where we can see that the particle runs fast when 'going down' the potential and it slows down as it moves up the steep slope of the potential.…”

“…In this section we derive the internal entropy production rateṠ i at stationarity [46][47][48][49], assuming that the observer is able to distinguish whether the RnT particle is in its right-or left-moving state. We discuss the case where the observer is not able to distinguish the RnT particle's state in appendix E. Other observables we calculate are in the appendix: mean square displacement (appendix F), two-point correlation function (appendix H), two-time correlation function (appendix I), expected velocity (appendix G) and stationary distribution (appendix D).…”

“…A positive entropy production rate is thus the signature of non-equilibrium and it indicates the breakdown of time-reversal symmetry. The entropy productionṠ i (t) of a drift-diffusive particle in free space with velocity w and diffusion constant D is known to beṠ i (t) = 1/(2t) + w 2 /D [46], where the first contribution is due to the relaxation to steady state and is independent of the system parameters, and the second contribution is due to the steady-state probability current.…”

Scaling of Entropy Production under Coarse Graining in Active Disordered Media

“…the extent to which their phenomenology differs from that of a collection of passive particles driven by a bath. The rate of entropy production [1,2] allows for such differentiation by capturing time-reversal symmetry breaking at the microscopic scale [3]. While the injection of energy ensures a strong departure from equilibrium at the single-agent level, these systems do not necessarily exhibit nonequilibrium features at larger spatio-temporal scales [4][5][6][7][8].…”

“…Entropy production and coarse graining. The starting point of our analysis is a Markovian jump process satisfying the master equation Ṗn (t) = m (P m (t)w m,n − P n (t)w n,m ) (1) for the probability P n (t), with w n,m the non-negative transition rate from state n to state m = n. The average rate of internal entropy production at steady-state is defined as [2] Ṡi = 1 2 n,m J n,m ln π n w n,m π m w m,n…”