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arXiv:1112.1876 · Ukrainian Journal of Physics

Quantum-Thermal Fluctuations of Effective Macroparameters and Their Correlations

Alexander D. Sukhanov, O. N. Golubjeva, V. G. Baryakhtar

Published Feb 6, 2022• 1 citation
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Abstract

The application of the conventional theory of macroparameter fluctuations has been shown to go beyond the framework of the thermodynamic description in a number of cases. The principles of the theory of quantum-thermal fluctuations of effectivemacroparameters and their correlations have been formulated. The theory satisfies the applicability conditions of equilibriumthermodynamics and is based on effective macroparameters, which take the integral stochastic action of the environment into account at any temperatures. The correlator of conjugate macroparameters, namely, the effective entropy and the effective temperature, has been calculated. The correlator was found to be proportional to the effective action which characterizes the stochastic environment. The pair correlators for the conjugate effective parameters "entropy–temperature" and "coordinate–momentum" have been demonstrated to depend linearly on the effective action, with their minimum values being determined by Planck's constant.

DOI
10.15407/ujpe56.10.1120

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The application of the conventional theory of macroparameter fluctuations has been shown to go beyond the framework of the thermodynamic description in a number of cases. The principles of the theory of quantum-thermal fluctuations of effectivemacroparameters and their correlations have been formulated. The theory satisfies the applicability conditions of equilibriumthermodynamics and is based on effective macroparameters, which take the integral stochastic action of the environment into account at any temperatures. The correlator of conjugate macroparameters, namely, the effective entropy and the effective temperature, has been calculated. The correlator was found to be proportional to the effective action which characterizes the stochastic environment. The pair correlators for the conjugate effective parameters "entropy–temperature" and "coordinate–momentum" have been demonstrated to depend linearly on the effective action, with their minimum values being determined by Planck's constant.
We propose an approach that makes it possible to narrow the gap between the macro- and micro-descriptions of nature (i.e., between thermodynamics and quantum mechanics). This approach is based on introducing the quantum-thermostat model in the form of cold and thermal vacuums. In this case we start from our previously developed microdescription in the form of ( ħ, k ) dynamics instead of the standard quantum mechanics. It is based on using the complex wave function of vacuum. We emphasize the universal role of the stochastic environmental influence in the appearance of correlations between the fluctuations of conjugate parameters on micro- and macrolevels at any temperatures.
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In thermal equilibrium, the fluctuation-dissipation theorem relates the linear response and correlation functions in a model and observable independent fashion. Out of equilibrium, these relations still hold if the equilibrium temperature is replaced by an observable and frequency-dependent parameter (effective temperature). When the system achieves a long-time thermal state all of these effective temperatures should be equal and constant. Following this approach we examine the long-times regime after a quantum quench in a system with bipartite entanglement in which the asymptotic values of the observable are compatible with the ones obtained in a Gibbs ensemble. We observe that when the initial entanglement is large, and for a large range of (intermediate) frequencies, the effective temperatures corresponding to the analyzed local and nonlocal operators approach an approximate constant value equal to the temperature that governs the decay of correlations. Still, the residual frequency dependence in the effective temperature, and the differences observed among observables, discards strict thermalization.
We suggest a more general than quantum statistical mechanics (QSMQSMQSM) microdescription of objects in a heat bath taken into account a vacuum as an object environment - modification of quantum mechanics at finite temperatures; we call it (ℏ,k)(\hbar, k)(ℏ,k)-dynamics ($ \hbar kD$). This approach allows us in a new manner to calculate some important macroparameters and to modify standard thermodynamics. We create an effective apparatus for features description of nearly perfect fluids in various mediums. As an essentially new model of an object environment we suppose a quantum heat bath and its properties, including cases of cold and warm vacuums, are studied. We describe the thermal equilibrium state in place of the traditional density operator in term of a wave function the amplitude and phase of which have temperature dependence. We introduce a new generative operator, Schroedingerian, or stochastic action operator, and show its fundamental role in the microdescription. We demonstrate that a new macroparameter, namely the effective action, can be obtained through averaging of the Schroedingerian over the temperature dependent wave function. It is established that such different parameters as internal energy, effective temperature, and effective entropy and their fluctuations can be expressed through a single quantity - the effective action.
Thermodynamics traditionally deals with macroscopic systems at thermal equilibrium. However, since the very beginning of the theory, its range of applicability has only increased, nowadays being applied to virtually every field of science, and to systems of extremely different size. This thesis is devoted to the study of thermodynamics in the quantum regime. It contains original results on topics that include: Work extraction from quantum systems, fluctuations of work, the energetic value of correlations and entanglement, and the thermodynamics of closed quantum many body systems. First, we study work extraction from thermally isolated systems. Here the notion of passive states naturally arises, as those quantum states from which no work can be extracted. We start by characterising the set of passive states, and find the most energetic passive states, a dual family to the well known Gibbs (or thermal) states. Remarkably, passive states have the property of activation: When considered as a whole, several copies of passive states can become nonpassive. We study the dynamics of activation processes, and find a relation between the entanglement generated and the speed of the process. Next, we consider the possibility of extracting work from a system using an auxiliary thermal bath. In this case, according to the second law of thermodynamics, the amount of work is bounded by the free energy difference. We develop corrections to this law which arise from the finite size and the structure of the bath. We go on by studying the fluctuations of work. Fluctuations are particularly relevant for small systems, where their relative size is comparable to the average value itself. However, characterising the fluctuations in the quantum regime is particularly difficult, as measurements generically disturb the state. In fact, we derive a no go result, showing that it is not possible to exactly measure the fluctuations of work in quantum coherent processes. Despite this result, we develop a new scheme that allows for their approximate measurement. An important part of this thesis is devoted to the relation between quantum correlations and work. We start by considering a set of correlated states which are thermal at the local level, in which case the extractable work can only come from the correlations. We compute the amount of work that can be stored in entangled, separable and correlated states with a fixed entropy, by finding the corresponding optimal states and protocols. These results provide fundamental bounds on the potential of different type of correlations for work storage and extraction. Next, we consider the converse scenario, and study the creation of correlations from thermal states. We find thresholds on the maximal temperature for the generation of entanglement. We also work out the minimal work cost of creating different types of correlations, including total correlations, entanglement, and genuine multipartite entanglement. Finally, we study the thermodynamics of closed quantum systems. Here we use one of the most important recent insights from the study of equilibration in quantum systems: Closed many body systems do not equilibrate, but can be effectively described as if they had equilibrated when looking at a restricted, physically relevant, class of observables. Importantly, the corresponding equilibrium state is not necessarily a Gibbs state, but may be very well given by a Generalized Gibbs ensemble state. With this in mind, we develop a framework for studying entropy production and work extraction in closed quantum systems. La termodinàmica va néixer com una teoria dedicada a l'estudi de cossos macroscòpics en equilibri tèrmic. 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This chapter is devoted to the classical limit of the quantum dissipative action obtained in Chapter 3. We show how it yields Langevin, Fokker–Planck and optimal path descriptions of classical stochastic systems. These approaches are used to discuss activation escape, fluctuation relation, reaction models and other examples. Classical dissipative action In Section 3.2 we derived the Keldysh action for a quantum particle coupled to an Ohmic environment, Eq. (3.20). If only linear terms in the quantum coordinate X q (t) are kept in this action, it leads to a classical Newtonian equation with a viscous friction force, Eq. (3.21). Such an approximation completely disregards any fluctuations: both quantum and classical . Our goal now is to do better than that and to keep classical thermal fluctuations, while still neglecting quantum effects. To this end it is convenient to restore the Planck constant ħ in the action and then take the limit ħ → 0. For dimensional reasons, the factor ħ -1 should stay in front of the entire action. To keep the part of the action responsible for the classical equation of motion (3.21) free from the Planck constant it is convenient to rescale the quantum component as X q → ħ X q . Indeed, when this is done all terms linear in X q do not contain ħ. Finally, to have the temperature in energy units, one needs to substitute T with T /ħ.