Anomalous Thermal Relaxation and the Mpemba Effect in Overdamped Langevin Systems and Markov Jump Processes

The Mpemba effect is an example of anomalous thermal relaxation. It occurs when a physical system cools down faster, starting from a hot temperature, than starting from a warm temperature when coupled to the same cold bath. There’s also an inter- esting counterpart in the heating process, often dubbed the Inverse Mpemba effect. The two effects have been observed in a variety of systems, including water, clathrate hydrates, polymers, magnetic alloys, quantum dots, quantum circuits, and colloidal systems. This work concerns our efforts in understanding the Mpemba effect, guided by the recent colloidal experiments, and it’s applications in Markov jump processes. Below, you will find a summary of our findings.
In particular, we will focus more on the Strong Mpemba effect – a more pronounced version of the Mpemba effect, characterized by a jump in the relaxation time, which yields an exponentially faster approach to equilibrium. Sometimes, the “regular” Mpemba effect is referred to as the Weak Mpemba effect to delineate between the two. In an attempt to gain intuition with the occurrence of the Mpemba effect, we model an overdamped Brownian particle diffusing on a potential energy landscape. We solve the Fokker-Planck equation for a piecewise constant potential for two wells separated by a barrier. Using this simple model that can be solved exactly, we ex- plore the phase space and note that we can observe the Mpemba effect in systems that lack a notion of metastability. This challenged the previous intuition that the Mpemba effect requires metastability. In the considered physical system, the borders of the areas where the effect happens correspond to either eigenvector changes of direction or to phase transitions. Finally, we discuss the topological aspects of the strong Mpemba effect and propose using topology to search for the Mpemba effect in a physical system. Later, colloidal experiments conducted measured the inverse Mpemba effect for a system without metastability [1].
Next, we now consider a continuous and smooth potential energy landscape like the one used by [2]. We derive the condition for the Mpemba effect in the small- diffusion limit of overdamped Langevin dynamics on a double-well potential. Our results show the strong Mpemba effect occurs when the probability of being in a well at initial and bath temperature match, which agrees with experiments. We also, for the very first time, derive the conditions for the weak Mpemba effect and express the conditions for the effects in terms of mean first passage times. We confirm our theoretical findings by simulating the dynamics using Monte Carlo simulations.
In the next two chapters, we study the Mpemba effect in Markov jump processes on linear reaction networks as a function of the relaxation dynamics. The dynamics are characterized by the so-called load factor, which is introduced to control the transition rates in a manner that ensures the system still relaxes to the same thermal equilibrium (i.e., detailed balance holds). We find a surprising connections between optimal transport and the Mpemba effect. Optimal transport is a resource-efficient way to transport the source distribution to a target distribution in a finite time. By “a resource-efficient way,” what is often meant, and what we will consider, is with the least amount of entropy production. Our paradigm for a continuum system is a particle diffusing on a potential landscape, while for a discrete system, we use a three- and four-state Markov jump process. The Mpemba effect is generically associ- ated with high entropy production in the continuous case. At large yet finite times, the system evolution toward the target is not optimal in this respect. However, in the discrete case, we show that for specific dynamics, the optimal transport and the strong variant of the Mpemba effect can occur for the same relaxation protocol.
In our last example, we provide analytical results and insights on when the Mpemba effect happens in the unimolecular reactions of three and four species as a function of the dynamics. We derive that in the unimolecular reactions of three species, the regions of the Strong Mpemba effect in cooling and heating are non- overlapping and that there is, at most, a single initial temperature leading to the Strong Mpemba effect. Next, we apply our results for the Markov jump processes on a Maxwell demon setup. Maxwell demon setups first appeared as thought experiments that explored the connection between information processing and thermodynamics. The first one to introduce such concepts was James Clerk Maxwell. We show that one can utilize the Strong Mpemba effect to have shorter cycles of the Maxwell demon device, leading to increased power output. We find a region of parameters where the device has increased power output and stable operation without sacrificing efficiency