The Mathematical Foundations of Quantum Thermodynamics

It is proposed a representation of the basic laws (i.e. the zeroth, first, second and third laws) in thermodynamics for quantum systems in the pure and mixed ensembles, respectively. We show that the basic laws are represented by parameters that specify respective quantum states. The parameters are the elements of the thermodynamic state space θ  and the state space ϑ  of the mixed ensemble for quantum systems. The introduction of such parameters is based on a probabilistic nature of quantum theory. Consistency between quantum theory and classical thermodynamics is preserved throughout the formulation for the representation of the thermodynamical laws in quantum systems (quantum thermodynamics). The present theory gives the mathematical foundations of quantum thermodynamics.


Introduction
Thermodynmics is a universal theory not only for classical but for quantum systems. Classical thermodynamics has been well established by different approaches [1] [2] [3]. Above all, the theoretical importance of thermodynamical consideration in quantum systems (quantum thermodynamics) is emphasized in textbooks [4] [5]. When we consider the thermodynamics for quantum systems, the most important is the change in entropy since entropy is a constant of motion under the unitary transformation generated by a system Hamiltonian [6] [7]. The internal energy of the system plays the same role as temperature [8]. It In subsection 2.3, we introduce an entropy function for the system of mixed states as a map on thermodynamic state space θ  and ϑ  for pure and mixed ensembles. We shall see that the entropy function ensures it is defined for all states in terms of θ 's ( θ ∈  ) and ϑ 's ( ϑ ∈  ). We will show that thermodynamic temperatures can be defined as a function of θ 's and ϑ 's for the system of mixed ensembles. In subsection 2.4, the third law of thermodynamics for the quantum systems of mixed states is briefly discussed. The absolute zero temperature is the state that the system is in a single quantum state. Finally, summary and concluding remarks are given in section 3.

Thermodynamics for Systems in Mixed Quantum States
In the previous paper [9], we have considered the case where thermodynamic quantum systems are in pure ensembles. The state of quantum systems in a pure ensemble is described by , and θ 's in θ  . It should be noticed that the origin of θ is a probabilistic nature of quantum mechanics but ϑ is due to the `classical' probability of occurrence j ω that the state vector is j Ψ . In this section, we derive a representation of zeroth, first, second, and third laws of thermodynamics for a quantum system in a mixed ensemble.
At first, we have to introduce the concept of an ensemble j w into the probability amplitude i a defined by , is complete orthogonal basis. We define the probability amplitude in a mixed ensemble as follows: The probability is thus given by Recall that 2 j i Ψ is the usual quantum mechanical probability taken with respect to the respective state i. Equation (2) tells us that these probabilities must further be weighted by the corresponding fractional populations j w . Notice how probabilistic concepts enter twice: first in In the following subsections, we derive the representation of the basic laws of quantum thermodynamics for a system in mixed ensembles.

The Zeroth Law of Quantum Thermodynamics
The zeroth law of quantum thermodynamics for a system in mixed ensembles is expressed by making use of the parameter ϑ 's in ϑ  . In order to represent the zeroth law, we need to introduce the Lemma L1' as follows: Now one can compare two quantum states in thermodynamic sense since those parameters θ 's and ϑ 's can be used to describe two or more systems being equivalent. This leads to the zeroth law of quantum thermodynamics. Let us prepare three systems, where superscripts indicate labels of respective systems. The zeroth law of quantum thermodynamics (equivalence relation among quantum states) is described by the following relation: Let us first consider the following relation for the system in mixed ensembles: If and , then , β ϑ ϑ ∈  respectively, and a symbol  here denotes that the state in the left-hand side is equivalent to the state in the right-hand side.
Proof of Equation (4) .  The zeroth law: We are now in a position to discuss some consequences obtained by introducing the parameters k θ and k ϑ to specify the corresponding thermodynamic states of the systems in mixed quantum ensembles. It is clear from Equation (4) that the transitive law holds:

The First Law of Quantum Thermodynamics
The first law of thermodynamics states that heat is a form of energy, and thermodynamic processes are therefore subject to the principle of conservation of energy, meaning that heat energy cannot be created or destroyed. In order to discuss the first law and obtain the representation of the first law of quantum thermodynamics, let us consider the internal energy of the system in mixed quantum ensembles. The internal energy m U of the system in mixed quantum states is given by the expectation value of Hamiltonian  : is the internal energy of the ensemble j.
By calculating the total differential of Equation (6), we obtain Analogous to the case for a system in a pure ensemble [9], we identify  (7) is reduced to is the case of a quantum system in a pure quantum ensemble since a pure quantum ensemble means  Distinguishing two kinds of transfer of energy, as heat and as thermodynamic work, adopted for thermodynamic processes, we prove that the equalities are true. Let us consider the small change in the outcome is given by where L is the work coordinate related to the work done on the system. Then A change in the internal energy of the system in a mixed ensemble is generally related to a force defined by so that Equation (10)  where the definite probability ij p can then be replaced by the probability function ( ) ( ) Thereby, from Equations (7) and (13)  ; .
is expressed in terms of those parameters θ and ϑ : ; .
We would like to note that Equation (15) reduces to ( )

The Second Law of Quantum Thermodynamics
In this subsection, we will give a definition of entropy to describe the entropy principle (viz., the second law of thermodynamics) for quantum system described by a mixed state. The entropy principle states that adiabatic accessibility of any two states is described by an entropy inequality. Here we should refer to the adiabatic process since the second law treated here is defined for the process. The process is characterized by 0 j d Q ′ = for mixed ensembles. This is ensured ; P x Q y θ ϑ remains unchanged throughout the process (see C1' and argument below). In other words, adiabatic process is a process such that ; P x Q y θ ϑ remains unchanged. It should be noted that adiabatic process allows to change a value of L since it affects only the work We note that this general definition for entropy can describe all types of entropy functions including well known Boltzmann, Gibbs, and Shannon entro-pies. The entropy m S defined by the map (16) is clearly a state quantity and ensures that m S can be defined for all states (i.e. pure and mixed ensembles) in terms of θ and ϑ .
In order to obtain a representation of the second law in terms of θ and ϑ for quantum system, it must be shown that determining parameters θ and ϑ .
In order to show this, let us show that m U is specified by θ 's, ϑ 's and L's . As in classical thermodynamics, heat in quantum system is also defined as a form of energy flow. Once the internal energy of a quantum system is well defined, heat is also well defined. Thus the following Lemma (L2') is established: S θ ϑ . This keeps consistency between an entropy function defined in the entropy principle (see Ref. [9], Section 2) and the statement (16) for a mixed ensemble of the system. We finally obtain a representation of the second law of quantum thermodynamics for a system in a mixed ensemble in terms of θ and ϑ :