Stochastic Thermodynamics of two-state Systems

Abstract

The central topic of this thesis will be the study of fluctuations that occur in small scale systems due to limited system size. We will focus on a Master equation approach, and on a two state system. We will choose such a two-state system because of its simplicity, and try to illustrate a few physical principles of much broader validity on this particular system. Despite the simplicity of this description, we believe it can serve as a starting point for more realistic models. It is however important to beware of accidental simplifications or symmetries that are obtained only because of the simple nature of the system (as shown in the second chapter). In a first chapter we introduce some basic notations and the main tools we will need further in the text. More specifically we will discuss some properties of stochastic matrices, some elements of traditional and stochastic thermodynamics, introduce the Master equation description and derive the fluctuation theorems in this context. We introduce generating functions, the long time limiting descriptions by means of the Cumulant Generating Function (CGF) and the Large Deviation Function (LDF) and the simplifications that come about when considering a piecewise constant driving. In the second chapter we will point out that for a certain (quite simple) system, we obtain the same symmetry that is usually ascribed to the Fluctuation Theorem (FT), but without any thermodynamic input at all (i.e. without assuming dynamic detailed balance in the individual rates imposed by each separate reservoir with respect to their own equilibrium distribution.) Because the fact that the symmetry holds even in those circumstances seems to be due to the system's simplicity rather then a physical principle, and the symmetry reduces to the FT in the case that detailed balance does hold, it seems caution is needed in associating symmetries of this type systematically to the FT. We also provide a more systematic way to take the Legendre transform in some cases without the need to verify the solutions (to discard the false one) afterwards. In this chapter we will also introduce the standard system that will be used in the applications throughout the text. We also present a way to calculate the Legendre transform for scaled cumulant generating functions (or just CGF's) of this type in a way that avoids discarding the unphysical solution and makes it's analyticity more manifest. We go on about the FT in the next chapter to discuss the circumstances under which the detailed form of these theorems allows for an enhanced interpretation in terms of entropy produced in original and conjugated experiment (i.e. an experiment where both driving and trajectory are modified in a prescribed way, to be specified later). We will show there that there is always at least one initial condition (that can be chosen to start the experiment from) for which this enhanced interpretation is valid (the 'echo state'), and determine this initial condition for each of the three detailed FTs. We calculate the limit cycle of the original piecewise driving and show that the required initial condition is just the time-reversed of this limit cycle. We also provide a way to correct the outcome of the experiment when one does not control the system sufficiently as to start the experiment from the right initial condition ('shadowing procedure'). Finally, in the last chapter we will use the framework of stochastic thermodynamics to study the statistics of the efficiency of a two state system functioning as a thermal machine (a refrigerator, heat pump or heat engine). More specifically we will consider the example of a refrigerator that transfers heat from a colder reservoir to a hotter one by modulating its energy levels in a (discontinuous) piecewise constant fashion and manipulating the coupling strength of its interaction with each of the two reservoirs synchronously in a different way for each reservoir (so they remain in 'anti-phase' with each other). This work started with the proposition to try to repeat what was done by G. Verley in [1] for the statistics of work in contact with a single reservoir to the case of two reservoirs at different reservoirs, and to involve also the exchanged heat with one of those reservoirs. We will count this exchanged heat and performed work by means of a joint generating function for which we calculate the long time behavior CGF exactly. We then numerically perform a Legendre transformation to obtain the Large Deviation function for the exchanged heat per cycle and the performed work per cycle. A so called contraction then gives us the large deviation function for the efficiency (COP for a fridge). We will also present an alternative way to obtain this LDF for the efficiency by performing a similar procedure directly on the joint CGF, which has the large advantage that the numerical errors due to the numerical Legendre transform are avoided, as I was able to calculate the joint CGF analytically but not its LDF. We perform this contraction again numerically for the general case and analytically in the close to equilibrium limit. We compare the results for the three main types of dynamics of the dynamics (Fermi,Bose, and Arrhenius rates) and the effect of varying each of the parameters while the others are held fixed for the far from equilibrium region (qualitatively) as well, and we consider also the long- and short period and close to equilibrium limits separately in the appendices. We end with some reections on the LDF for efficiency (variance? and exact P), and point out a series of possibly interesting things that deserve further attention. We also calculate the exact (i.e. far from equilibrium) average uxes and efficiency, divide the parameter space in different modes of operation, after which we focus usually on the case of the refrigerator (the motor being treated in [2], except for a few important figures and results which are also restated for/translated the other cases. These exact fluxes, as well as the expanded second cumulants (with exception of the variance for heat Cqq) are calculated in the appendices. There we derive a key formula for the trace of the propagator on which this whole last chapter rests, expand the (trace of the) CGF (first for work alone, in powers of a) and then the cumulants for the joint CGF, in powers of β and a up to second order. For Cqq we give the zero-th order analytically (as it is the only cumulant that does not vanish at equilibrium). Also the long and short period limits are calculated in the appendices. At the end of the last chapter finally, there is a brief discussion of the most obvious features in the results for the LDF of the refrigerator COP, and a comparison with the close to equilibrium predictions is made. Also, in the last section, a number of important results (the more general cases for driving that is not invariant up to a phase under time reversal, and the case of tight coupling) that were excluded in the bulk of the text, are mentioned briefly, and a proof that the maximum of the LDF of COP lies at Carnot efficiency is provided that is valid arbitrarily far from equilibrium, with references to the publications where these are treated in more detail, together with other model systems that were not discussed here.

Ik zal in deze tekst aan de hand van een Markov proces in een systeem met slechts twee mogelijke toestanden dat beschreven wordt door middel van een Master vergelijking trachten enkele van de mogelijkheden van de zogenaamde stochastische thermodynamica te illustreren. Vaak zullen hierbij concepten gebruikt worden die ook in een bredere context hun betekenis behouden maar die specfiek voor deze twee-toestandssystemen makkelijker analytisch oplosbaar zijn terwijl dat anders meestal niet mogelijk is. We zullen op sommige plaatsen echter ook aandacht besteden aan zaken die louter het gevolg zijn van deze overgesimplficeerde beschrijving.