## Abstract

Classical formulations of the entropy concept and its interpretation are introduced. This is to motivate the definition of the quantum von Neumann entropy. Some general properties of quantum entropy are developed, such as the quantum entropy which always increases. The current state of the area that includes thermodynamics and quantum mechanics is reviewed. This interaction shall be critical for the development of nonequilibrium thermodynamics. The Jarzynski inequality is developed in two separate but related ways. The nature of irreversibility and its role in physics are considered as well. Finally, a specific quantum spin model is defined and is studied in such a way as to illustrate many of the subjects that have appeared.

### Keywords

- classical
- quantum
- partition function
- temperatures
- entropy
- irreversible

## 1. Introduction

The laws of thermodynamics are fundamental to the present understanding of nature [1, 2]. It is not surprising then to find they have a very wide range of applications beyond their original scope, such as to gravitation. The analogy between properties of black holes and thermodynamics could be extended to a complete correspondence, since a black hole in free space had been shown to radiate thermally with a temperature

In the twentieth century, it became clear that the microworld was described by a different kind of physics along with mathematical ideas that need not be taken into account in describing the macroworld. This is the subject of quantum mechanics. Even though the new quantum equations have similar symmetry properties as their classical counterparts, it also reveals numerous phenomena that can contribute at this level to the problems mentioned above. These physical phenomena which play various roles include the phenomenon of quantum entanglement, the effect of decoherence in general, and the theory of measurements as well.

The purpose of this is to study the subject of entropy as it applies to quantum mechanics [8, 9]. Its definition is to be relevant to very small systems at the atomic and molecular level. Its relationship to entropies known at other scales can be examined. It is also important to relate this information from this new area of physics to the older and more established theories of thermodynamics and statistical physics [10, 11, 12, 13, 14, 15]. To summarize, many good reasons dictate that the arrow of time is specified by the direction of increase of the Boltzmann entropy, the von Neumann macroscopic entropy. To relate the quantum Boltzmann approach to irreversibility to measurement theory, the measuring apparatus must be included as a part of the closed quantum mechanical system.

## 2. Entropy and quantum mechanics

Boltzmann’s great insight was to connect the second law of thermodynamics with phase space volume. This he did by making the observation that for a dilute gas,

Clearly, the macrostate

This

The approach of Gibbs, which concentrates primarily on probability distributions or ensembles, is conceptually different from Boltzmann’s. The entropy of Gibbs for a microstate

In (2),

Then clearly

The probability density for the system in the equilibrium macrostate

From the standpoint of mathematics, these expressions for classical entropies can be unified under the heading of the Boltzmann-Shannon-Gibbs entropy [16]. A very general form of entropy which includes those mentioned can be defined in a mathematically rigorous way. To do so, let

when the integrand is integrable.

This includes the classical Boltzmann-Gibbs entropy when

It also includes the Shannon entropy appearing in information theory in which

In this case, (5) gives the entropy to be

In attempting to translate these considerations to the quantum domain, it is immediately clear that a perfect analogy does not exist.

Although the situation is in many ways similar in quantum mechanics, it is not identical. The irreversible incompressible flow in phase space is replaced by the unitary evolution of wave functions in Hilbert space and velocity reversal of

This formula was given by von Neumann. It generalizes the classical expression of Boltzmann and Gibbs to the realm of quantum mechanics. The density matrix with maximal entropy is the Gibbs state. The range of

Here,

and

von Neumann justifies (10) by noting that

for

and

A correspondence can be made between the partitioning of classical phase space

where

Note the difference that in the classical case, the state of the system is described by

### 2.1 Properties of entropy functions

Entropy functions have a number of characteristic properties which should be briefly described in the quantum case. The set of observables will be the bounded, self-adjoint operators with discrete spectra in a Hilbert space. The set of normal states can be taken to be the density operators or positive operators of trace one.

The entropy functional satisfies the following inequalities. Let

with equality if all

Subadditivity holds with equality if and only if

and

where the first equality holds iff

The formal expression will be interpreted as follows. If

Concavity of the function

which gives (15) and (16). If

This is to say that

The concept of irreversibility is clearly going to be relevant to the subject at hand, so some thoughts related to it will be given periodically in what follows. A possible way to account for irreversibility in a closed system in nature is by the various types of course-graining. There are also strong reasons to suggest the arrow of time is provided by the direction of increase of the quantum form of the Boltzmann entropy. The measuring apparatus should be included as part of the closed quantum mechanical system in order to relate the quantum Boltzmann approach to irreversibility to the concept of a measurement. Let

There is an amplification process of the

and each element of the abelian subalgebra of

Define the projection operators:

Suppose

This is a mixture of states in each of which

The transformation

where

The following definition can now be stated. A history is said to *decohere* if and only if

A state is called decoherent with respect to the set of

This implies that

The relative or conditional entropy between two states

When

The last two inequalities are known as joint concavity and monotonicity of the relative entropy. The following result may be thought of as a quantum version of the second law.

**Theorem**: Suppose the initial density matrix is decoherent at zero time (29) with respect to

and it is not an equilibrium state of the system. Let

where **(e)**

Proof: Set

The first equality uses the cyclic property of the trace and the definition of **e**) follows from (32).

Of course, entropy growth as in the theorem is not necessarily monotonic in the time variable. For this reason, it is usual to refer to fixed initial and final states. For thermal systems, a natural choice of the final state is the equilibrium state of the system. It is the case in thermodynamics that irreversibility is manifested as a monotonic increase in the entropy. Thermodynamic entropy, it is thought, is related to the entropy of the states defined in both classical and quantum theory. Under an automorphic time evolution, the entropy is conserved. One application of an environment is to account for an increase. A type of course-graining becomes necessary together with the right conditions on the initial state to account for the arrow of time. In quantum mechanics, the course-graining seems to be necessary and may be thought of as a restriction of the algebra and can also be interpreted as leaving out unobservable quantum correlations. This may, for example, correspond to decoherence effects important in quantum measurements. Competing effects arise such as the fact that correlations becoming unobservable may lead to entropy increase. There is also the effect that a decrease in entropy might be due to nonautomorphic processes. Although both effects lead to irreversibility, they are not cooperative but rather contrary to one another. The observation that the second law does hold implies these nonautomorphic events must be rare in comparison with time scales relevant to thermodynamics.

## 3. Quantum mechanics and nonequilibrium thermodynamics

Some aspects of equilibrium thermodynamics are examined by considering an isothermal process. Since it is a quasistatic process, it may be decomposed into a sequence of infinitesimal processes. Assume initially the system has a Hamiltonian

The average external energy

When the parameter

Each instantaneous infinitesimal process can be broken down into a part which is the work performed; the second is the heat transformed as the system relaxes to equilibrium. This breakup motivates us to define

so

By integrating over the infinitesimal segments, we find

Inverting Eq. (38) for

Substituting into the relation for

It remains to study

By the chain rule

So

This relation only holds for infinitesimal processes. For finite and irreversible processes, there may be additional terms to the entropy change. This has been quite successful at describing many different types of physical system [17, 18, 19].

A deep insight has come recently into the properties of nonequilibrium thermodynamics which could be achieved by regarding work as a random variable. For example, consider a process in which a piston is used to compress a gas in a cylinder. Due to the nature of the gas and its chaotic motion, each time the piston is pressed, the gas molecules exert a back reaction with a different force. This means the work needed to achieve a given compression changes each time something is carried out.

Usually a knowledge of nonequilibrium processes is restricted to inequalities such as the Jarzynski inequality. He was able to show by interpreting work

Suppose the system is always prepared in the same state initially. A process is carried out and the total work

Jarzynski showed that the statistical average of

where

In macroscopic systems, individual measurements are usually very close to the average by the law of large numbers. For mictoscopic systems, this is usually not true. In fact, the individual realizations of

To get (50) requires detailed knowledge of the system’s dynamics, be it classical, quantum, unitary, or whatever.

Consider nonunitary quantum dynamics. Initially, the system has Hamiltonian

Immediately after this measurement,

where

The Hamiltonian is

No heat has been exchanged with the environment, so any change in the environment has to be attributed to the work performed by the external agent and is

where both

To get an expression for

where

And some over all allowed events, weighted by their probabilities, and arrange the terms according to the values

This has the inverse Fourier transform

Using (55), we obtain that

Hence, it may be concluded that

This turns out to be somewhat easier to work with than

A formula for the quantum mechanical formula for the moments can be found as well. The average work is

Using

Nothing has been assumed about the speed of this process. Thus inequality (50) must hold for a process arbitrarily far from equilibrium.

## 4. Heat flow from environment approach

There is another somewhat different way in which the Jarzynski inequality can be generalized to quantum dynamics. In a classical system, the energy of the system can be continuously measured as well as the flow of heat and work. Continuous measurement is not possible in quantum mechanics without disrupting the dynamics of the system [20].

A more satisfactory approach is to realize that although work cannot be continuously measured, the heat flow from the environment can be measured. To this end, the system of interest is divided into a system of interest and a thermal bath. The ambient environment is large, and it rapidly decoheres and remains at thermal equilibrium, uncorrelated and unentangled with the system. Consequently, we can measure the change in energy of the bath

For a system that has equilibrated with Hamiltonian

Conversely, any operator-sum represents a complete positive superoperator. The set of operators

The interest here is in the dynamics of a quantum system governed by a time-dependent Hamiltonian weakly coupled to an extended, thermal environment. Let the total Hamiltonian be

where

By following the unitary dynamics of the combined total system for a finite time and measuring the final state of the environment, a quantum operator description of the system dynamics can also be obtained:

Here

and

Suppose the environment is large, with a characteristic relaxation time short compared with the bath-system interactions, and the system-bath coupling

The Hermitian operator of a von Neumann-type measurement can be broken up into a set of eigenvalues

The state of the system after this interaction is

The result of the measurement can be represented by using a Hermitian map superoperator

An operator-value sum maps Hermitian operators into Hermitian operators:

In the other direction, any Hermitian map has an operator-value-mean representation. Hermitian maps provide a particularly concise and convenient representation of sequential measurements and correlation functions. For example, suppose Hermitian map

The correlation function

It may be shown that just as every Hermitian operator represents some measurement on the Hilbert space of pure states, every Hermitian map can be associated with some measurement on the Liouville space of mixed states.

A Hermitian map representation of heat flow can now be constructed under assumptions that the bath and system Hamiltonian are constant during the measurement and the bath-system coupling is very small. A measurement on the total system is constructed, and thus the bath degrees of freedom are projected out. This leaves a Hermitian map superoperator that acts on the system density matrix alone. Let us describe the measurement process and mathematical formulation together.

Begin with a composite system which consists of the bath, initially in thermal equilibrium weakly coupled to the system:

Measure the initial energy eigenstate of the bath so based on (76):

Now allow the system to evolve together with the bath for some time:

Finally, measure the final energy eigenstate of the bath:

Taking the trace over the bath degrees of freedom produces the final normalized system density matrix where trace over

Replace the heat bath Hamiltonian by

Collecting the terms acting on the bath and system separately and replacing the Krauss operators describing the reduced dynamics of the system, the result is

To summarize, it has been found that the average Boltzmann weighted heat flow is represented by

where

The paired Hermitian map superoperators act at the start and end of a time interval. They give a measure of the change in the energy of the system over that interval. This procedure does not disturb the system beyond that already incurred by coupling the system to the environment. The Jarzynski inequality now follows by applying this Hermitian map and quantum formalism. Discretize the experimental time into a series of discrete intervals labeled by an integer

The system Hamiltonian is fixed within each interval. It changes only in discrete jumps at the boundaries. The heat flow can be measured by wrapping the superoperator time evolution of each time interval

In (85),

This product actually telescopes due to the structure of the energy change Hermitian map (84) and the equilibrium density matrix (65). This leaves only the free energy difference between the initial and final equilibrium ensembles, as can be seen by writing out the first few terms

In the limit in which the time intervals are reduced to zero, the inequality can be expressed in the continuous Lindblad form:

## 5. A model quantum spin system

A magnetic resonance experiment can be used to illustrate how these ideas can be applied in practice. A sample of noninteracting spin-

In units where

If we set

and

The work segment is implemented by introducing a very small field of amplitude

Typically,

The oscillating field plays the role of a perturbation which although weak may initiate transitions between the up and down spin states and will be most frequent at the resonance condition

The time evolution operator

Substituting (43) into the evolution equation for

It is found that

Using the commutation relations of the Pauli matrices and the fact that

it is found that the terms in the evolution equation can be simplified

By means of these results, it remains to simplify

Taking these results to (95), we arrive at

This means

and the full-time evolution operator is given by

Since the operators

To express (100) otherwise, suppose

Now

Since

Consequently, (100) can be used to prove that

Since

the evolution operator is then given by

The functions

Apart from a phase factor, the final result depends only on

This expression represents the transition probability per unit time a transition will occur. Since the unitarity condition

Now that

If

Substituting

Consider the average work. Suppose

The average work at time

since

The equilibrium free energy is

This is a consequence of the fact that

Given the matrices for

Set

Substituting (119) into (118), we can conclude

This is the Jarzynski inequality, since it is the case that

From (121), the first and second moments can be obtained; for example

As a consequence, the variance of the work can be determined

A final calculation that may be considered is the full distribution of work

Using the Fourier integral form of the delta function, (124) can be written as

Work taken as a random variable can take three values

The second law would have us think that

The work performed by an external magnetic field on a single spin-

## 6. Conclusions

We have tried to give an introduction to this frontier area that lies in between that of thermodynamics and quantum mechanics in such a way as to be comprehensible. There are many other areas of investigation presently which have had interesting repercussions for this area as well. There is a growing awareness that entanglement facilitates reaching equilibrium [21, 22, 23]. It is then worth mentioning that the ideas of einselection and entanglement with the environment can lead to a time-independent equilibrium in an individual quantum system and statistical mechanics can be done without ensembles. However, there is really a lot of work yet to be done in these blossoming areas and will be left for possible future expositions.