For pt.I see ibid., vol.2, p.203 (1981). In a thermodynamic system the following quantities are intimately related: 'thermodynamic probability' W, entropy S(U) as a function of internal energy U, U itself, and the variance sigma E 2 of the energy. A further close relation with the energy spectrum (E n ) of the system is pointed out here. Once the purely mathematical relations between these concepts are understood, it is shown how only one general case needs to be considered in order to follow Planck's and Einstein's arguments concerning quantisation. A choice of the entropy function S(U), for example, determines (E n ). The results clarify the interrelationships of the assumptions.
For pt.I see ibid., vol.2, p.203 (1981). In a thermodynamic system the following quantities are intimately related: 'thermodynamic probability' W, entropy S(U) as a function of internal energy U, U itself, and the variance sigma E 2 of the energy. A further close relation with the energy spectrum (E n ) of the system is pointed out here. Once the purely mathematical relations between these concepts are understood, it is shown how only one general case needs to be considered in order to follow Planck's and Einstein's arguments concerning quantisation. A choice of the entropy function S(U), for example, determines (E n ). The results clarify the interrelationships of the assumptions.
Statistical formulations of thermodynamic entropy, such as those by Boltzmann and Gibbs, were originally developed for classical systems and are well understood in that context. However, the foundational aspects of quantum statistical mechanics remain an area of active debate and are yet to be fully understood. This work is motivated by the need to develop a comprehensive understanding of the statistical measures of thermodynamic entropy in quantum systems - a topic intimately connected to the phenomenon of quantum thermalization. In particular, we investigate the conditions under which the von Neumann entropy can be regarded as a valid statistical measure of thermodynamic entropy in quantum systems. This paper demonstrates that the equivalence between the von Neumann and thermodynamic entropies is not universal, but instead depends on several subtle and often overlooked assumptions. In this context, we also briefly revisit key criticisms of von Neumann entropy - particularly its time-invariance and subadditivity - and argue that these concerns can be meaningfully addressed in the setting of thermodynamic systems. To substantiate some arguments and to clarify some issues, we provide suitable numerical results from our analysis of a spin-1/2 system.
Shannon’s entropy: -k Sum over i P(i) ln(P(i)) is expressed in terms of P(i) the probability for state i. If entropy exists as a concept separated from the first law of thermodynamics with dWork=0, then it is not clear what set of states i should be taken in quantum mechanics. In classical statistical mechanics, each particle is automatically described by its position and momentum vectors for no internal motion. For example a pure energy eigenstate wavefunction Wn(x) may be written in terms of many different orthonormal mathematical basis sets e.g. Wn(x)= Sum over i bi phi(i)(x) with probabilities bi*bi. Thus one may create many types of entropy expressions using Shannon’s form without any other physical considerations. For example, W(x)=Sum over p a(p)exp(ipx) and one may create Sp which is usually done in the literature. In previous notes we have argued that Sn = Sp(n)+Sx(n). For a particle in a box with infinite potential walls, Sx(n)=Sx. The sum of the two entropies ensures that the parameter L (box size) vanishes so that a quantum adiabatic transformation is isentropic. Thus a consideration other than Shannon’s form has been used to obtain Sn=Sp(n)+Sx for a particle in a box. Furthermore one may argue Shannon’s form even for a pure state based on S=S1+S2 where 1 and 2 represent two systems. A question still remains: Why are Sx and Sp entropies used and not some other set? Furthermore for a system with several energy levels ei one might ask why -Sum over i P(ei)ln(P(ei)) is used as in (1)? In a previous note (2) we considered that Shannon’s form of entropy follows from dE= TdS (dWork=0) with dE=Sum over i ei dP(ei). Furthermore we argued that P(ei) = C(T)exp(-ei/T) is a solution, but so is C(T) exp(-ei/T) exp(Si) where Si=Sx,i + Sp,i for the pure bound state. In other words, specific states related to the physical situation are chosen and entropy constructed from dE=TdS to reflect this. In this note we argue that entropy may be less related to counting arrangements than to considering the issues involved in energy transfers within a system.
Historically, thermodynamics was formulated for macroscopic systems followed by the development of statistical mechanics which provided microscopic formulations for quantities such as entropy. Entropy is traditionally linked to the ln of the number states available or to the Shannon’s form -k Sum over i P(ei) ln(P(ei)) where P(ei) is the probability for the quantity ei. In addition a variable T (linked to average kinetic energy) appears in both theories. Later the Schrodinger equation was developed to describe bound electron states in an atom etc. These states were characterized by a math object W(x) called the wavefunction and a set of discrete energy levels was calculated which matched data of emitted photons. In a previous note (1) we argued that the average energy En for a quantum bound state plays the same role of characterizing equilibrium as T in classical thermodynamics/statistical mechanics. We further argued that En is linked to parameters in the system. For example, for a particle in a box with infinite potential walls, En is linked to L as well as to n the energy level. One may change En by either changing n or L separately (or both). Holding L constant and changing n leads to a certain type of energy change. We argue this is similar to holding the volume of a gas constant and increasing T. Alternatively, one may leave n unchanged (a quantum adiabatic process) and change L i.e. a mechanical change or work. We argue that one may already consider the concept of the first law of thermodynamics at the quantum bound state level i.e. dE = dQ - dWork where Work is linked to a change in a physical parameter, such as L the length of the box, whereas dQ is linked to energy changes not involving a physical parameter e.g. a change in n the energy level. The Schrodinger equation yields solutions for W(x)=Sum over p a(p)exp(ipx) the wavefunction as well as En’s. We have argued in previous notes that W(x) is part of a conditional probability P(p/x)= a(p)exp(ipx)/W(x) and that En is constant at each point x just as T is constant at each x for classical statistical equilibrium (1). Thus a kind of statistical mechanics exists at the wavefunction level we argue. This statistical mechanics makes use of a free particle wavefunction exp(ipx) and weights a(p), but does not contain the usual P(x), P(p) picture of the classical world. We have shown in (2) that one may obtain P(x)=W*(x)W(x) and P(p)=a*(p)a(p) from the conditional probability. The essential physics we argue, however, is at the level of the wavefunction together with its own unique statistical scenario of exp(ipx)’s interacting with exp(ikx)’s from V(x)=Sum over Vkexp(ikx) (see (1)). Given that P(x) and P(p) exist for a quantum bound state, it may be convenient to use them in formulas because they are linked to a classical statistical picture. We argue that one may use P(x) and P(p) to construct an object S such that T dS does not change when one changes L, the size of box with infinite potential walls. We argue that such an object follows from properties of the Schrodinger equation together with various assumptions. We arrive at S=Sp+Sx where Sp=-k Integral dp a(p)a(p) ln{|a(p)a(p)}| and Sx= -k Integral dx W(x)W(x) ln(W(x)W(x)) for a given energy level. We do not use arguments related to taking ln of the number of states present or Shannon’s entropy. Rather we look for a solution which does not depend on L and is additive. We note, however, that for both W(x)W(x) and a(p)a(p) the solution found is mathematically consistent with the entropy form found by considering independent probabilities. This, however, applies to P(x) and P(p), but we argue that the physical situation is based on W(x) and a(p) as solutions of the Schrodinger equation. Nevertheless P(x) and P(p) are physical observables and not simply math constructs and so may think of them in a somewhat classical manner in terms of probability. Thus we argue that thermodynamics is already present at the level of a quantum bound state and that one may obtain Shannon’s entropy at this level using P(x) and P(p), but without considering the overall probability of independent probabilities as in information theory or numbers of states. Rather obtaining a function which is additive in Sp and Sx and eliminates L is the main consideration. We also argue that S=Sp+Sx carries over into mixed quantum state wavefunctions even though it is usually ignored. Thus we try to argue that it is the bound quantum state which gives rise to the notion of thermodynamics and entropy.
To understand in detail the relation between unitary quantum theory that describes our world at the microscopic scale and thermodynamics, which was long believed only to apply to macroscopic objects, is one of the most interesting and long-standing problems in physics. Recently, this problem has received renewed attention, in particular from the community of quantum information theory, but also from the field of statistical mechanics, inspired from stochastic thermodynamics. These results suggest that thermodynamics is also relevant for individual quantum systems, provided that they can be brought into contact with thermal baths. In this thesis, I use recently developed tools to provide new results both on fundamental questions, but also on questions which are of practical relevance for potential miniaturized thermal machines. In terms of fundamental questions, the results in this thesis contribute to understanding how the basic laws of thermodynamics and statistical mechanics can be understood directly from unitary quantum mechanics. In particular, I discuss and answer the following questions: i) How can we quantify the third law of thermodynamics using information theoretic methods? ii) How can we quantify the "thermodynamic value" of a state-transition in quantum systems? iii) How can we axiomatically characterize the non-equilibrium free energy and relative entropy? iv) How can we justify statistical ensembles from an operational perspective, without having to introduce either some probability measures or an information theoretic entropy measure? v) How can we understand the equilibration of closed quantum systems, how long does it take and how difficult is it to avoid? In the second part of the thesis I discuss in detail how experimental restrictions, which become important at the quantum scale, influence the ultimate thermodynamic bounds for thermal machines. In particular, the results in this thesis provide thermodynamic bounds on work-extraction and efficiencies of thermal machines in situations where 1.) an experimenter only has limited field strengths available, 2.) an experimenter cannot control the interactions between particles, but external fields arbitrarily well, and 3.) situations in which a small quantum system can only be strongly coupled to heat baths. These bounds are tight and I provide explicit examples illustrating the different behaviours. Finally, I come back to a classic problem in statistical physics: The emergence of spontaneous symmetry breaking. Here, I provide general and rigorous new results that show how symmetry-breaking stationary states emerge from fluctuations in order parameters in dissipative lattice models.
For pt.I see Found. Phys., vol.9, no.11-12, p819-29 (1979). A general assumption made in a previous paper is modified by the elimination of the metrical entropy in favour of empirical entropy. Without reference to absolute temperature or metrical entropy, one then arrives at a generic form of the canonical probability in phase in which temperature does not occur. It is argued that this is as it should be if one looks upon the canonical ensemble as merely being the representative of a system in thermal equilibrium with its surroundings. One motivation for this work is to take a view of 'statistical mechanics' which does away with probabilistic notions altogether.
For pt.I see ibid., vol.25, p.359 (1992). Regarding the discussion of Peres (1989) and Weinberg (1989) concerning a suitable definition of entropy in nonlinear quantum mechanics, two further observations are made. Firstly, by regarding the covariance matrix of the probability distribution over the phase space of wavefunctions as the nonlinear counterpart (p NL ) of the (linear) density matrix (p L ) and employing -Tr p NL In p NL , one obtains a limiting transition (as nonlinearities vanish), in which this entropy measure converges to the definition in ordinary quantum mechanics, -Tr p L In p L . Secondly, it is argued that Peres' contention that 'nonlinear variants of the Schrodinger equation violate the second law of thermodynamics' is flawed in that it relies upon the entropy of mixing of nonorthogonal states which as Dieks an van Dijk (1988) have indicated is an undefined concept. A proper approach to associating a quantum mechanical entropy with a mixture of a particle into two non-orthogonal states-by first estimating a suitable two-particle density matrix (p) and then employing -Tr p In p-is outlined.