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arXiv:1208.3948

Path Entropy Changes in Adiabatic Approximation

Jang-il Sohn

Published Aug 20, 2012• 0 citations
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Abstract

By applying adiabatic theorem to a Markovian system, we calculate the adiabatic and diabatic entropy changes along a path. As well known, the total path entropy change is separated into two parts, system and environment entropy changes, $ΔS_{tot} = ΔS_{sys} + ΔS_{env}$. The environment entropy change, $ΔS_{env}$, is divided again into two parts, an adiabatic contribution due to work, $ΔS_{\mathcal{W}}$, and a diabatic contributions due to heat, $ΔS_{\mathcal{Q}}$. In an adiabatic process, total path entropy change is same with the adiabatic path entropy change, $ΔS_{A}$, which is given by sum of system entropy change and adiabatic contribution, $ΔS_{A} = ΔS_{sys} + ΔS_{\mathcal{W}}$. Mathematical form of $ΔS_{A}$ is a type of excess heat entropy change, but $ΔS_{A}$ is due to work. By which, it is shown that the terms adiabatic and non-adiabatic contributions of $ΔS_{na}$ and $ΔS_{a}$ in [Phys. Rev. Lett. {\bf 104}, 090601 (2010)] should be completely switched, $i.e.$ $ΔS_{na} \rightarrow ΔS_{A}$ and $ΔS_{a} \rightarrow ΔS_{\mathcal{Q}}$ in fact.

DOI
10.48550/arxiv.1208.3948

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By applying adiabatic theorem to a Markovian system, we calculate the adiabatic and diabatic entropy changes along a path. As well known, the total path entropy change is separated into two parts, system and environment entropy changes, ΔStot=ΔSsys+ΔSenvΔS_{tot} = ΔS_{sys} + ΔS_{env}ΔStot​=ΔSsys​+ΔSenv​. The environment entropy change, ΔSenvΔS_{env}ΔSenv​, is divided again into two parts, an adiabatic contribution due to work, ΔSWΔS_{\mathcal{W}}ΔSW​, and a diabatic contributions due to heat, ΔSQΔS_{\mathcal{Q}}ΔSQ​. In an adiabatic process, total path entropy change is same with the adiabatic path entropy change, ΔSAΔS_{A}ΔSA​, which is given by sum of system entropy change and adiabatic contribution, ΔSA=ΔSsys+ΔSWΔS_{A} = ΔS_{sys} + ΔS_{\mathcal{W}}ΔSA​=ΔSsys​+ΔSW​. Mathematical form of ΔSAΔS_{A}ΔSA​ is a type of excess heat entropy change, but ΔSAΔS_{A}ΔSA​ is due to work. By which, it is shown that the terms adiabatic and non-adiabatic contributions of ΔSnaΔS_{na}ΔSna​ and ΔSaΔS_{a}ΔSa​ in [Phys. Rev. Lett. {\bf 104}, 090601 (2010)] should be completely switched, i.e.i.e.i.e. ΔSna→ΔSAΔS_{na} \rightarrow ΔS_{A}ΔSna​→ΔSA​ and ΔSa→ΔSQΔS_{a} \rightarrow ΔS_{\mathcal{Q}}ΔSa​→ΔSQ​ in fact.
The quantum adiabatic theorem is a widely applied result in quantum mechanics. However, recently, counterexamples were produced, indicating that the transition probabilities in some systems do not converge as predicted by the adiabatic theorem [K. P. Marzlin and B. C. Sanders, Phys. Rev. Lett. 93, 160408 (2004)]. In this paper, we examine a proof of the adiabatic theorem, paying specific attention to the validity of the assumptions made in the proof. We show that the convergence of the transition probabilities is path dependent when the change in the Hamiltonian is large. Finally, we consider the application of this reformulated form of the adiabatic theorem to the adiabatic evolution paradigm in quantum computation.
We analyze the production of entropy along nonequilibrium processes in quantum systems coupled to generic environments.First, we show that the entropy production due to final measurements and the loss of correlations obeys a fluctuation theorem in detailed and integral forms.Second, we discuss the decomposition of the entropy production into two positive contributions, adiabatic and nonadiabatic, based on the existence of invariant states of the local dynamics.Fluctuation theorems for both contributions hold only for evolutions verifying a specific condition of quantum origin.We illustrate our results with three relevant examples of quantum thermodynamic processes far from equilibrium.
Given the evolution of an arbitrary open quantum system, we formulate a general and unambiguous method to separate the internal energy change of the system into an entropy-related contribution and a part causing no entropy change, identified as heat and work, respectively. We also demonstrate that heat and work admit geometric and dynamical descriptions by developing a universal dynamical equation for the given trajectory of the system. The dissipative and coherent parts of this equation contribute exclusively to heat and work, where the specific role of a work contribution from a counterdiabatic drive is underlined. Next we define an expression for the irreversible entropy production of the system which does not have explicit dependence on the properties of the ambient environment; rather, it depends on a set of the system's observables excluding its Hamiltonian and is independent of internal energy change. We illustrate our results with three examples.
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.
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.)
Addendum Feb. 3, 2022 We restrict S=entropy=Sx+Sp to pure energy eigenstates. I.e. Wn(x) used in Sx is an energy eigenstate. In (1) we suggested the following form for the entropy of a quantum bound state: S = Integral dx W(x)W(x) ln[WW] + Integral dp a(p)a(p) ln[a(p)a(p)] ((1)) where W(x)= Sum over p a(p) exp(ipx). In principle this may be extended to nonpure bound states. In (2) we showed that this definition (which follows from a Shannon’s entropy with a particular choice of probability) ensures that an adiabatic change in the size of box with an infinite potential leads to no change in entropy as ln(1/L) from the first term of ((1)) cancels with ln(L) from the second for every energy level n. This seems to have some thermodynamic consequences. We revisit the particle in a box with infinite potential and also examine the quantum oscillator solution as the spring constant or mass is varied and find that it behaves in a similar way for all energy levels under an adiabatic transformation. We finally consider under what conditions such a transformation would lead to entropy changes.