Anomalous Thermal Relaxation in 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. An analogous effect exists in heating and is sometimes referred to as the Inverse Mpemba effect. The two effects have been observed in a variety of systems, including water, clathrate hydrates, polymers, magnetic alloys, quantum dots, and colloidal systems. This thesis discusses several theoretical advances in understanding the Mpemba effect on Markov jump processes and its relation to relaxation dynamics, information exchanges, and optimal transport. The paragraphs below detail our work.

We study the Mpemba effect in Markov jump processes on linear reaction networks as a function of the relaxation dynamics. A jump process is a random event where a system transitions between discrete states at fixed or random time intervals. Markov jump processes are discrete time-step processes where the outcome at any given time depends only on the configuration of the system at that time, i.e., it is independent of any past the system has gone through. 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 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. In particular, we focus on the Strong Mpemba effect – a more potent variant of the Mpemba effect, characterized by a jump in the relaxation time, which yields an exponentially faster relaxation. Sometimes, the "regular" Mpemba effect is referred to as the Weak Mpemba effect to delineate between the two. 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.

We then study the 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-state Markov jump process. The Mpemba effect is generically associated 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.

Subsequently, we study the effect of the topology of a network on the existence and extent of the Mpemba effect. For a system with quenched disorder and Metropolis-Hasting dynamics, we study how the thermal relaxation changes in response to rank-one modifications of the transition rates. We solve for the spectra of two modified matrices exactly. One of the modifications introduces the (weak) Mpemba effect to the system, while the other introduces the strong variant of the Mpemba effect.