Representation of the Basic Laws of Thermodynamics in Quantum Mechanics ()

We propose a representation of the basic laws, namely the zeroth, first, second and third law, in quantum thermodynamics. The zeroth law is represented by some parameters () that specify respective quantum states. The parameters are the elements of thermodynamic state space. The introduction of such parameters is based on a probabilistic nature of quantum theory. A quantum analog of the first law can be established by utilizing these parameters. The notion of heat in quantum systems is clarified from the probabilistic point of view in quantum theory. The representation of the second law can be naturally described in terms of these parameters introduced for the respective quantum systems. In obtaining the representation of quantum thermodynamics, consistency between quantum theory and classical thermodynamics should have been preserved throughout our formulation of quantum thermodynamics. After establishing the representation of the second law, the third law is discussed briefly. The relationship between thermodynamic temperatures and the parameters in is also discussed.

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Cite this paper

Suzuki, A. and Taira, H. (2018) Representation of the Basic Laws of Thermodynamics in Quantum Mechanics. *Journal of Modern Physics*, **9**, 2420-2436. doi: 10.4236/jmp.2018.914155.

1. Introduction

Thermodynamics is one of the theories which have high universality since thermodynamics as itself has been unchanged even if we now have a well-developed quantum theory. Classical thermodynamics has been well established by different approaches [1] [2] [3] . Lieb and Yngvason made the mathematical structure transparent by axiomatic approach [3] . Thermodynamics is a theory not only for classical but for quantum systems. Above all, the theoretical importance of thermodynamical consideration in quantum systems (quantum thermodynamics) is emphasized in text books [4] [5] . When we consider the thermodynamics for quantum systems, the important is the change in entropy since entropy is a constant of motion under the unitary transformation generated by a system Hamiltonian [6] [7] . A quantum heat engine has been investigated theoretically [8] [9] . Bender et al. studied a quantum Carnot cycle by considering a single quantum mechanical particle confined in a quantum well [9] . In their study, they found that the efficiency is equal to that of the Carnot cycle for classical case and proposed that the internal energy $U$ plays the same role as temperature. It should be however mentioned that in quantum system one cannot describe thermodynamic equibria in terms of a parameter like a temperature as in classical system [3] .

This paper deals with the following questions that must be answered: Can thermodynamical laws refer to the variation of states of a system represented by the quantum states such as those states (eigenstates) of the Hamiltonian for a single quantum mechanical particle confined in a quantum well? If thermodynamical laws exist in quantum thermodynamic systems, how can they be expressed? To answer these questions, we need a representation which connects thermodynamic states and quantum states.

In classical thermodynamics, states are represented by points on a state space. A typical example of the state space is just a collection of P (pressure) and V (volume), i.e., a P-V plane. In quantum mechanics, what space should be used in order to describe thermodynamic states for quantum systems? In quantum mechanics, quantum (pure) states are expressed by the elements of a complex Hilbert space $\mathcal{H}$. However, the Hilbert space itself does not play the same role as the state space in classical thermodynamics since comparing one state vector with the others in $\mathcal{H}$ must be done by comparing the components of each vector. Thus we start with introducing a set ${\mathcal{M}}_{\Psi}$ of the state vectors in $\mathcal{H}$ in order to obtain a suitable set which plays the same role as the state space in classical thermodynamics. That is, we introduce a set ${\mathcal{M}}_{\Theta}$ which plays the same role as the state space and derive a correspondence between ${\mathcal{M}}_{\Psi}$ and ${\mathcal{M}}_{\Theta}$. After that, we will show that the first law of quantum thermodynamics can be described by the elements in ${\mathcal{M}}_{\Theta}$ and the internal energy of a quantum system can be described as a function on ${\mathcal{M}}_{\Theta}$. We will also discuss the relation between those parameters Θ’s in ${\mathcal{M}}_{\Theta}$ and thermodynamic temperatures.

The first law of thermodynamics is a law of conservation of energy and states the equivalence of heat and work. We will discuss the equivalence between work and heat in quantum thermodynamics. We assume that the energy of a system (i.e., the internal energy) is given by the expectation value of the Hamiltonian $H$ : $U=\langle H\rangle ={\displaystyle {\sum}_{i}}{p}_{i}{E}_{i}$. In this expression, ${E}_{i}$ is the outcome of the expected energy state corresponding to a definite probability ${p}_{i}$ in a specified maximal test. Indeed, this probabilistic nature of quantum system plays a key role to establish a representation of the first law. Differentiating $U$ formally, we obtain the expression, $dU={\displaystyle {\sum}_{i}}\left({E}_{i}d{p}_{i}+{p}_{i}d{E}_{i}\right)$. The first term ${\sum}_{i}}{E}_{i}d{p}_{i$ implies there exists a non-mechanical source that induces a change in the internal energy of the system since a change in quantum states is in general determined by the unitary operator which does not change the definite probability (i.e., $d{p}_{i}=0$ ). The second term ${\sum}_{i}}{p}_{i}d{E}_{i$ implies a mechanical source that induces a change in the internal energy since we can trace the origin of $d{E}_{i}$ to an external parameter. As will be shown in Subsec. 3.2, the following identifications, ${d}^{\prime}Q={\displaystyle {\sum}_{i}}{E}_{i}d{p}_{i}$ and ${d}^{\prime}W={\displaystyle {\sum}_{i}}{p}_{i}d{E}_{i}$, are justified and are ensured by the existence of respective parameters $\Theta $ and $L$. We will show that the internal energy of quantum system is generally expressed in terms of parameters $\Theta $ and $L$, respectively, describing the equivalence relation among quantum states and external parameters. Therefore, the first law of quantum thermodynamics can be uniquely represented by these parameters. Once establishing the representation of the first law, it is worth to investigate the remaining thermodynamical laws (the second and third laws) for a quantum system described by quantum states.

In this paper, we propose a representation of the thermodynamical laws for quantum system in terms of the respective parameters and develop a theory of quantum thermodynamics based on the axiomatic theory of classical thermodynamics by Lieb and Yngvason [3] . In their formulation, the second law refers to the possible adiabatic transition of any two states in a state space.

This paper is organized as follows. In the next section, we present a brief review of classical thermodynamics. In Sec. 3, we state the basic notion of our formulation of quantum thermodynamics, and introduce a thermodynamic state space ${\mathcal{M}}_{\Theta}$ and a quantum state space ${\mathcal{M}}_{\Psi}$ and discuss the connection between them. In Subsec. 3.1 we show the existence of the zeroth law of quantum thermodynamics in the state space ${\mathcal{M}}_{\Theta}$. In Subsec. 3.2, the first law of thermodynamics and an adiabatic process are discussed. In Subsec. 3.3, we define entropy and give a representation of the second law, and discuss a relation among the adiabatic transitions, entropy and the term ${d}^{\prime}Q$. We refer to the third law in Subsec. 3.4. Finally, we give the results and discussion in Sec. 4.

2. Classical Thermodynamics

There are few approaches in thermodynamics [1] [2] [3] . Lieb and Yngvason’s approach is helpful to understand the logical structure of thermodynamics. If thermodynamical laws exist in quantum systems as well as in classical systems, there must be the same logical structure in both systems. According to their formulation, a structure of adiabatic accessibility on a state space (thermodynamic state space) is characterized by an entropy inequality, i.e., the second law of thermodynamics. In this section we present a brief review of classical thermodynamics due to Lieb and Yngvason [3] . Thermodynamics is a theory which discusses a transition between equilibrium states. The second law refers to the feasible transitions in adiabatic process.

We start with introducing a formulation of the axiomatic thermodynamics proposed by Lieb and Yngvason [3] . In their formulation, the second law of thermodynamics is represented by the entropy principle.

Entropy principle: There is a real-valued function on all states of all systems (including compound systems), called entropy and denoted by $S$. Entropy has the following properties:

・ Monotonicity : When
$X$ and
$Y$ are comparable states^{1}, then

$X\prec Y\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{if}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{only}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}S\left(X\right)\le S\left(Y\right)\text{\hspace{0.05em}}\mathrm{.}$ (1)

・ Additivity : If $X$ and $Y$ are states of some (possibly different) systems and if $\left(X\mathrm{,}Y\right)$ denotes the corresponding state in the composition of the two systems, then the entropy is additive for these states, i.e.,

$S\left(X,Y\right)=S\left(X\right)+S\left(Y\right)\text{\hspace{0.05em}}.$ (2)

・ Extensivity : $S$ is extensive, i.e., for each $t>0$ and for each state $X$ and its scaled copy $tX$,

$S\left(tX\right)=tS\left(X\right)\text{\hspace{0.05em}}.$ (3)

It should be noted that entropy is determined by the physical (or thermodynamic) state of the system. In the entropy principle, $X$ and $Y$ (e.g., energy and volume) describe equilibrium states and are the elements of a state space (denoted by $\Gamma $ ). A system is then represented by the state space $\Gamma $ on which a relation “ $\prec $ ” of adiabatic accessibility is defined. The definition of adiabatic accessibility is as follows:

Adiabatic accessibility: A process whose only effect on the surroundings is exchange of energy with a mechanical source. This means that as a state arrives at new one, a state of surroundings is the same as before, in other words, the device returns to its initial state at the end of the process.

Lieb and Yngvason [3] proved that existence and uniqueness of entropy are equivalent to certain simple properties of a relation “ $\prec $ ” (A1~A6) and a comparison hypothesis (Ch):

A1. Reflexivity : $X\stackrel{\text{A}}{~}X$.

A2. Transitivity : $X\prec Y$ and $Y\prec Z$ implies $X\prec Z$.

A3. Consistency : $X\prec {X}^{\prime}$ and $Y\prec {Y}^{\prime}$ implies $\left(X\mathrm{,}Y\right)\prec \left({X}^{\prime}\mathrm{,}{Y}^{\prime}\right)$.

A4. Scaling invariance : If $X\prec Y$, then $tX\prec tY$ for $\forall \text{\hspace{0.05em}}t>0$.

A5. Splitting and Recombination : For $0<t<1$, $X\stackrel{\text{A}}{~}\left(tX\mathrm{,}\left(1-t\right)X\right)$.

A6. Stability : If, for some pair of states, X and Y, $\left(X\mathrm{,}\u03f5{Z}_{0}\right)\prec \left(Y\mathrm{,}\u03f5{Z}_{1}\right)$ holds for a sequence of ò’s tendency to zero and some states ${Z}_{0}$ and ${Z}_{1}$, then $X\prec Y$.

Ch. Comparison hypothesis : The Ch holds for a state space $\Gamma $ if any two states $X$ and $Y$ on the space are comparable states, i.e., $X\prec Y$ or $Y\prec X$.

In the axiom A1, the symbol $\stackrel{\text{A}}{~}$ denotes that two states $X$ and $Y$ are adiabatic equivalent; It describes a situation where both of the relations, $X\prec Y$ and $Y\prec X$, hold. It should be noted that the axioms, A3, A5 and A6, are defined on the product of state space $\Gamma \times \Gamma $, where $\left(X\mathrm{,}Y\right)\in \Gamma \times \Gamma $. The Ch asserts that any two states on the same state space are comparable. Generally, the structure on the state space $\Gamma $ is determined by the axioms (A1~A6) and the comparison hypothesis (Ch) under the condition of adiabatic accessibility.

Let us consider a meaning of the entropy principle. Let $X\mathrm{,}{X}^{\prime}\mathrm{,}Y\mathrm{,}{Y}^{\prime}\mathrm{,}\cdots $ be the elements of the state space $\Gamma $. Imagine that we have a list of all possible pairs of states $X$, $Y$ such that $Y$ is adiabatic accessible from $X$. The foundation of thermodynamics and the essence of the second law are that this list, $X$ and $Y$, such as $X\prec Y$, can be simply encoded by the entropy function $S$ defined on a set of all states of systems (including compound systems). This means that $Y$ is adiabatic accessible from $X$, i.e., $X\prec Y$ if and only if $S\left(X\right)\le S\left(Y\right)$ (entropy inequality). The entropy function should be kept consistency with the structure of the state space $\Gamma $ characterized by A1~A6 and Ch. Thus, we can characterize the structure based on the definition of adiabatic accessibility on the state space $\Gamma $ by using the entropy inequality. Combining the axioms (A1~A6) and the Equation (2), one can describe the entropy principle for systems including a compound system.

Let us consider a compound system in which $X$, ${X}^{\prime}$ and $Y$, ${Y}^{\prime}$ are the states of system A and system B, respectively. In this case, the entropy principle is mathematically expressed as follows:

$\left(X\mathrm{,}Y\right)\prec \left({X}^{\prime}\mathrm{,}{Y}^{\prime}\right)\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{if}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{only}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}S\left(X\right)+S\left(Y\right)\le S\left({X}^{\prime}\right)+S\left({Y}^{\prime}\right)\mathrm{.}$ (4)

Note that all states
$\left({X}^{\prime}\mathrm{,}{Y}^{\prime}\right)$ such that
$X\prec {X}^{\prime}$ and
$Y\prec {Y}^{\prime}$ are adiabatically accessible from

$\left(X\mathrm{,}Y\right)$. It is then important to notice that
$\left({X}^{\prime}\mathrm{,}{Y}^{\prime}\right)$ can be adiabatically accessible from
$\left(X\mathrm{,}Y\right)$ even if
${X}^{\prime}$ is not adiabatically accessible from
$X$. In such a case, entropy increase,
$S\left({Y}^{\prime}\right)-S\left(Y\right)$, in the process compensates for a loss,
$S\left({X}^{\prime}\right)-S\left(X\right)$, so as to satisfy the statement (4). Therefore, the inequality,
$S\left(X\right)+S\left(Y\right)\le S\left({X}^{\prime}\right)+S\left({Y}^{\prime}\right)$, characterizes the possible adiabatic transitions for the compound system even when
$S\left(X\right)\ge S\left({X}^{\prime}\right)$. It means that it is sufficient to know the entropy of each part of the compound system in order to decide which transition is feasible due to the interactions between the two subsystems.

For later use we write the entropy principle [the statement (4)] in terms of $U$ and $V$, where $U$, $V$ denote the internal energy and the volume of a system, respectively. Putting $X=\left({U}_{\text{A}}\mathrm{,}{V}_{\text{A}}\right)$ and $Y=\left({U}_{\text{B}}\mathrm{,}{V}_{\text{B}}\right)$, one obtains from the statement (4):

$\begin{array}{l}\left(\left({U}_{\text{A}}\mathrm{,}{V}_{\text{A}}\right)\mathrm{,}\left({U}_{\text{B}}\mathrm{,}{V}_{\text{B}}\right)\right)\prec \left(\left({{U}^{\prime}}_{\text{A}}\mathrm{,}{{V}^{\prime}}_{\text{A}}\right)\mathrm{,}\left({{U}^{\prime}}_{\text{B}}\mathrm{,}{{V}^{\prime}}_{\text{B}}\right)\right)\\ \text{\hspace{0.05em}}\text{if}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{only}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}S\left({U}_{\text{A}}\mathrm{,}{V}_{\text{A}}\right)+S\left({U}_{\text{B}}\mathrm{,}{V}_{\text{B}}\right)\le S\left({{U}^{\prime}}_{\text{A}}\mathrm{,}{{V}^{\prime}}_{\text{A}}\right)+S\left({{U}^{\prime}}_{\text{B}}\mathrm{,}{{V}^{\prime}}_{\text{B}}\right)\text{\hspace{0.05em}}\mathrm{.}\end{array}$ (5)

It should be noted that the state of the compound system composed of system A and system B is described by $\left({U}_{\text{A}}\mathrm{,}{U}_{\text{B}}\right)$ only in the case where the volume is invariant during the process. The statement (5) makes sense in the case where $X$ is an extensive variable. However, there exists a particular case in which $X$ is an intensive variable.

One of the aims in this paper is to obtain a representation of entropy inequality for quantum system corresponding to the statement (4).

3. Quantum Thermodynamics

In order to obtain the representation of the zeroth, first, second, and third laws for quantum thermodynamics, we have to introduce a state space in order to describe thermodynamic states of quantum system. In the previous section, we have seen that the thermodynamic states of classical system denoted by capital Roman letters, $X\mathrm{,}Y\mathrm{,}Z$, etc. defined as the elements of state space $\Gamma $ satisfy certain simple properties of the relation “ $\prec $ “ (A1~A6) and the comparison hypothesis (Ch). From the mathematical point of view, we expect that the thermodynamic structure of quantum thermodynamics should also have the same structure as that of classical thermodynamics.

In order to develop a representation of quantum thermodynamics, we must introduce a thermodynamic state space for quantum system since in quantum theory, quantum system is described by the complex Hilbert space $\mathcal{H}$ and the states of quantum system are in general described by the elements in $\mathcal{H}$ : $|{\Psi}_{\alpha}\rangle $, $|{\Psi}_{\beta}\rangle $, etc. Here, Greek letter (subscript) denotes the label of respective states of quantum system. In the following, we use a symbol