We improve on our version of the second law of thermodynamics as a deterministic theorem for quantum spin systems [ 1 ] in two basic aspects. The first concerns the general statement of the second law: spontaneous changes in an adiabatically closed system will always be in the direction of increasing (mean) entropy, which rises to a maximal value. In order to arrive at this statement, we have to extend the previous definition of adiabatic transformation, in order to include sudden interactions, and thereby generalize the “barrier model”. The cornerstone of the second law is then seen to be the fact that the dynamics may induce a basic structural transition between states of infinite systems in the limit of large times, which, physically, represent times much larger than specific relaxation times. Two specific examples concern the transition from pure to mixed states in two different universality classes of dynamics in one dimension, one being the exponential model, which has exactly soluble dynamics and does not display a phase transition, the other the Dyson model(s), for which neither the dynamics nor the statistical mechanics are exactly soluble, and which do exhibit a (ferromagnetic) phase transition, first demonstrated by Dyson. It is also shown, as a consequence of results of Albert and Kiessling on the Cloitre function, that there is strong graphical evidence that the dynamics of the Dyson models are chaotic for large times, and, thereby, that the mechanisms of approach to equilibrium differ for these two universality classes in a profound way, which depend exclusively on the dynamics, and not on the states and observables.
We improve on our version of the second law of thermodynamics as a deterministic theorem for quantum spin systems [ 1 ] in two basic aspects. The first concerns the general statement of the second law: spontaneous changes in an adiabatically closed system will always be in the direction of increasing (mean) entropy, which rises to a maximal value. In order to arrive at this statement, we have to extend the previous definition of adiabatic transformation, in order to include sudden interactions, and thereby generalize the “barrier model”. The cornerstone of the second law is then seen to be the fact that the dynamics may induce a basic structural transition between states of infinite systems in the limit of large times, which, physically, represent times much larger than specific relaxation times. Two specific examples concern the transition from pure to mixed states in two different universality classes of dynamics in one dimension, one being the exponential model, which has exactly soluble dynamics and does not display a phase transition, the other the Dyson model(s), for which neither the dynamics nor the statistical mechanics are exactly soluble, and which do exhibit a (ferromagnetic) phase transition, first demonstrated by Dyson. It is also shown, as a consequence of results of Albert and Kiessling on the Cloitre function, that there is strong graphical evidence that the dynamics of the Dyson models are chaotic for large times, and, thereby, that the mechanisms of approach to equilibrium differ for these two universality classes in a profound way, which depend exclusively on the dynamics, and not on the states and observables.
We improve on our version of the second law of thermodynamics as a deterministic theorem for quantum spin systems in two basic aspects. The first concerns the general statement of the second law: spontaneous changes in an adiabatically closed system will always be in the direction of increasing mean entropy, which rises to a maximal value. Two specific examples concern the transition from pure to mixed states in two different universality classes of dynamics in one dimension, one being the exponential model, the other the Dyson model, the dynamics of the latter exhibiting strong graphical evidence of quantum chaos, as a consequence of the results of Albert and Kiessling on the Cloitre function.
The second law of thermodynamics for adiabatic operations -- constraints on state transitions in closed systems under external control -- is one of the fundamental principles of thermodynamics. On the other hand, it is recently established that even pure quantum states can represent thermal equilibrium. However, pure quantum states do not satisfy the second law in that they are not passive, i.e., work can be extracted from them if arbitrary unitary operations are allowed. It therefore remains unresolved how quantum mechanics can be reconciled with thermodynamics. Here, based on our key quantum-mechanical notions of thermal equilibrium and adiabatic operations, we address the emergence of the second law for adiabatic operations in the thermodynamics limit. We first introduce infinite-observable macroscopic thermal equilibrium (iMATE); a quantum state, including pure states, is in iMATE if the expectation values of all additive observables agree with their equilibrium values. We also introduce a macroscopic operation as unitary evolution generated by a time-dependent additive Hamiltonian, which is regarded as corresponding to adiabatic operations. Employing these concepts, we show that no extensive work can be extracted from any quantum state in iMATE through any macroscopic operations. Furthermore, we introduce a quantum-mechanical form of entropy density such that it agrees with thermodynamic entropy density for any quantum state in iMATE. We then prove that for any initial state in iMATE, this entropy density cannot be decreased by any macroscopic operations, followed by a time-independent relaxation process. Our theory thus proves two different forms of the second law, by adopting macroscopically reasonable classes of observables, equilibrium states, and operations. We also discuss the time scales of macroscopic operations in these results.
A consistent theory of non-equilibrium thermodynamics for Markovian open quantum systems has been developed in the late seventies in analogy with Classical Irreversible Thermodynamics. The time-evolution of these open systems is usually described by means of effective master equations in Lindblad form, that turn out to be reliable when there is a separation of time-scales between system and environment, such that memory effects are negligible and the so-called Markovian approximation is justified. In this framework, the variations of energy and entropy in the system are consistently described, distinguishing between heat and work contributions and providing a statement of the second law of thermodynamics as positivity of the entropy production.\nHowever, there is empirical evidence that many physical systems like photosynthetic complexes,\nopto-mechanical resonators and superconducting qubits, just to mention a few, experience more general non-Markovian dynamics. The formulation of the laws of thermodynamics in a non-Markovian setting is matter of current research and represents the main topic of the present work. As a first step, we show that the entropy production defined as in the Markovian case can be negative for a class of non-Markovian dynamics. We argue that this outcome should not be interpreted as a violation of the second law of thermodynamics because the environment must be explicitly taken into account in the balance of entropy in the non-Markovian setting. In order to justify this claim we adopt a more general point of view, studying a closed bipartite quantum system, such that each of the two subsystems plays the role of a finite environment out of equilibrium for the other one. We concentrate on the balance of energy first and construct an effective Hamiltonian for each subsystem using physically reasonable requirements; then we define heat and work as in the standard Markovian treatment, with the effective Hamiltonian replacing the free Hamiltonian. It turns out that, in our framework, the work power is perfectly balanced between subsystems, while the correlations can store a part of energy locally inaccessible and exchange it with both subsystems in the form of heat.\nConcerning the balance of entropy, a quite general formulation of the second law of thermodynamics can be given as follows: under the assumption of a factorized initial state for the compound system, the sum of the total variations of the entropies in the two subsystems is always nonnegative. We show with an explicit example that this general formulation does not correspond to the statement presented in the framework of Markovian master equations, which should not be considered a priori the second law of thermodynamics.\nIn the last part of the thesis we concentrate of the so-called fluctuation relations, that are results extending the thermodynamic formalism beyond the behavior of average quantities. After reviewing the main theoretical outcomes, such as the Jarzynski equality and the Crooks fluctuation theorem, we describe a proposal to access experimentally the work performed on an ensemble of diatomic molecules by a time-dependent electric field coupled with their vibrational degree of freedom. This procedure could then be used to test the quantum Jarzynski equality. With respect to the results so far appeared in the literature, in which the left-hand side of the equality is inferred from an experiment and the right-hand side is computed according to a model, in our proposed setting we should be able to estimate from the experiment both the left-hand side and the right-hand side of the equality, independently.
Deriving the laws of thermodynamics from an underlying microscopic picture is a central quest of statistical mechanics. The present article focuses on the derivation of the first and second law of thermodynamics for closed and open quantum systems, where such foundational questions in addition become practically relevant for emergent nanotechnologies. Recent progress is presented in a self-contained way, with an emphasis on general ideas and useful tools instead of particular applications. We identify limits and conceptual shortcomings of previous approaches. Then, we propose a novel, unifying perspective, which starts from a microscopic definition of nonequilibrium thermodynamic entropy. The change of this entropy is identified with the entropy production, which satisfies a fluctuation theorem for a large class of initial states. Moreover, this entropy production can be naturally separated into a quantum and a classical component. Within our framework, we introduce the notions of recoverable work and remaining heat. In the case of a weakly perturbed ideal thermal bath this approach is in quantitative agreement with previous ones, thus ensuring their thermodynamic consistency.
In Part I we argued that both the first law of thermodynamics dE=TdS-dW and the form S/(-k) = Integral dp dx P(x)P(p) ln{P(x)P(p)} = Integral P(x)ln(P(x)) + Integral P(p) ln(P(p)) may be established at the level of a pure quantum bound state with P(x)=W(x)W(x) and P(p)=a(p)a(p) where W(x)=wavefunction = Sum over p a(p)exp(ipx). We further argued that S=Sp+Sx in order that L, the length of a box with infinite potential walls, should disappear (from S) so that a quantum adiabatic transformation where n=energy level remains constant is isentropic. In fact, we argued that the P(x)ln(P(x)) form follows from this rule and an assumption of additive entropies Sp and Sx. As a result, probability at the pure bound state level is not based on the maximization of entropy as P(x)=W(x)W(x) and P(p) =a(p)a(p) and W(x) follows from the solution of the time-independent Schrodinger equation. In this note we examine the case of two level bound state system (analyzed in (1)) to see how the thermodynamic temperature behaves in such a case and examine the maximization of entropy. T is already present in the pure energy state picture (TdS), but for two levels or more, it plays a different role in that P(ei) where ei is the ith energy level becomes a function of T which enters as a Lagrange multiple during the maximization of S with respect to P(ei) with the constraint Sum over i eiP(ei) = Eave. No such maximization of entropy occurs at the pure bound state level. We note, however, that for a quantum adiabatic transformation for a particle in a box with infinite potential walls, P(ei) values must remain constant as L changes. Ei is proportional to 1/LL so a solution is to use ei/T with T is proportional to 1/LL. Thus the parameter T already enters P(ei) in order to ensure that an adiabatic transformation is isentropic. In the pure bound state, one was forced to use P(x)P(p)ln(P(x)P(p)) because P(x) is proportional to 1/L and P(p) to L. In other words in the pure state there are two kinds of probabilities (p and x) and one must ensure that L disappears. For the two level system there is only one type of probability which contains ei(L) so this L must be offset within the probability itself i.e. by using T. Thus thermodynamical and statistical mechanical behaviours which seem to be present at the pure quantum bound state level have different features for multiple level systems even though the first law of thermodynamics remains the same as does the form of entropy. What seems to change is the role of T temperature. We try to examine these features in this note. In particular we also find that maximization of entropy subject to constant energy which is usually linked to high N scenarios (i.e. factorial calculation in the high N limit) follows from the first law of thermodynamics with dW=0. Thus it holds even for one quantum particle with two energy levels (particle in a box with infinite walls). This, however, makes use of the form E=Sum over i ei P(ei). Such a form does not hold for example for a pure bound state.
Historically statistical mechanics was developed for gasses for which there are very large numbers of molecules (i.e. on the order of 10 to 23rd power). As a result factorial schemes were developed to count the number of arrangements possible, the ln taken of this result and Stirling’s approximation for large N applied. The Shannon’s entropy form S/(-k) = Sum over i P(ei)ln(P(ei)) is consistent with this large N approach. Furthermore, large N statistical mechanics was linked to thermodynamics, in particular the first law dE=TdS - PdV. Surprisingly then, statistical mechanical results were applied to systems with small numbers of particles (e.g. a nucleus with about 100 nucleons etc). In Parts I and II we argued that the first law of thermodynamics describes two separate kinds of changes associated with energy changes. The second, work, is linked to the change of a mechanical parameter such as the volume of a box etc. The energy of a bound quantum particle in a box with an infinite potential is proportional to nn/LL. Thus we argued that a pure quantum bound state may also be associated with the first law of thermodynamics with changes in L with n, the energy level, held constant representing pure work. We further argued that for a bound state two classical like probabilities exists P(x)=W(x)W(x) and P(p)=a(p)a(p) where W(x)=wavefunction = Sum over p a(p)exp(ipx) with P(x) being proportional to 1/L and P(p) to L. For a quantum adiabatic transformation (i.e. changing L with n staying constant) the term dS=0. If one wishes to create S from P(x) and P(p) and desires linearity i.e. S=Sp(n)+Sx, then Sp=Integral dp P(p)ln(P(p)) and Sx= Integral dx P(x) ln(P(x)) is a solution. If one next considers a quantum wavefunction for two levels W(x,t) = a(1)exp(-i E1 t) W1(x) + a(2) exp(-iE2 t) W2(x) with Eave = Sum over n en P(en) where Wn=a(n)a(n), one may again utilize the first law of thermodynamics. In Parts I and II we carried over the entropy form P(ei)ln(P(ei)). In this note we wish to argue that this form follows at the two energy level picture by considerations of a function which consists entirely of probabilities i.e. P(ei) and is linear in the sense that such a function for two identical two state systems should yield S1+S2 as the first law should apply to each system. Furthermore P(ei) should be a function of P(ei/T) where T is a parameter which behaves as 1/LL because ei is proportional to 1/LL. In this way any function of P(ei) remains unchanged under a quantum adiabatic transformation for which i’s stay constant, but L changes. As a result, the entropy at the quantum level seems to follow from such arguments and not from large N approximations of factorial counting approaches. In large N situations it is argued that P(ei)=C(T)exp(-ei/T) follows from maximizing S with respect to the form of P(ei) subject to constant average energy. For small N situations (e.g. the 2 level example) one may solve Sum over i ei d(a(i)a(i)) = T dS directly without maximization. The maximization approach follows directly from dE=TdS (work=0) because dE= Sum ei dP(ei) in such scenarios. This is of the form of a Lagrange constraint This has been pointed out in Part II. In this note, we argue that the maximization of entropy is not needed at all. For dE=Sum over i ei dP(ei) and the ln form of entropy P(ei) is the inverse function of ln. Thus we argue that linearity and not large N arguments are enough to obtain Shannon’s form of entropy and that this ln form suggests P(ei/T) = C(T)exp(-ei/T) if dE=Sum over ei dP(ei)= TdS (for dW=0). Thus maximization of entropy subject to constant average energy is also not needed (although it does follow mathematically from dE=TdS.)