Boltzmann entropy of a freely expanding quantum ideal gas

Understanding the emergence of the second law for macroscopic systems from the reversible microscopic dynamics was the remarkable achievement of Boltzmann boltzmann1897 ; feynman2017character ; lanford1976derivation ; penrose1990emperor ; greene2004fabric ; lebowitz1993macroscopic ; Lebowitz_PT1993 ; Goldstein_PD2004 ; griffiths1994 . He constructed an entropy function, satisfying the second law, for a typical individual microstate $X$ of a macroscopic system that is in a macrostate specified by the values of a collection of macrovariables. This Boltzmann entropy is defined for systems both in and out of equilibrium as the logarithm of the “number” of microstates corresponding to the system’s macrostate. For a classical system, the microstates are specified by points $X$ in phase space, and the Boltzmann entropy of $X$ is, up to an additive constant, the logarithm of the phase space volume of the set of all microstates for which the macrovariables have the same values as they do for $X$. For a quantum system, the microstates are specified by wave functions $|\Phi\rangle$. This difference, which allows observables to have indeterminate values in the quantum microstate, can make the precise definition of quantum macrovariables and macrostates somewhat subtle goldstein2017 ; tasaki2016 ; mori2018 ; de2006 , and also allows for new non-classical choices of macrovariables, as we illustrate below. The Boltzmann entropy of $|\Phi\rangle$ is then the logarithm of the number of independent wave functions that have the same values of the chosen macrovariables as $|\Phi\rangle$ does. At thermal equilibrium, for all proper choices of macrovariables this definition coincides, in its extensive part, with the thermodynamic entropy of Clausius callen1998 .

In a previous paper some of the authors investigated the time evolution of the Boltzmann entropy for a freely expanding classical ideal gas, using two different choices of macrovariables chakraborti2021entropy . In the present paper, we extend this to a freely expanding quantum ideal gas. We provide numerical results for the evolution of the microstates and the associated entropy functions during free expansion. We consider both fermions and bosons and present results in the low and high-temperature regimes.

Before moving on, we mention some of the earlier works relevant to our study. Quantum quench of non-interacting fermions under various protocols have been studied using Wigner functions in dean2018fermions ; manas2018 . Lattice fermions evolving from domain wall initial conditions have been studied with a focus on the evolution of the density profile and the growth of entanglement entropy eisler2013 ; scopa2021 . The diagonal entropy for isolated quantum systems has been studied rigol2011 for both integrable and non-integrable cases. Some other recent relevant discussions of entropy in quantum systems in the nonequilibrium setting can be found in Refs. deutsch2019a ; deutsch2019b ; solano2016theory .

The rest of the paper is organized as follows. In Sec. (2) we discuss ideas related to the definition of macrovariables, macrostates and the Boltzmann entropy in quantum systems. In Sec. (3) we introduce the specific example that we will be analyzing in the paper. In Sec. (4) we define our model of a quantum ideal gas and discuss aspects of its microscopic evolution. In particular, we discuss properties of the single-particle density matrix in Sec. (4.1) and the Wigner distribution function in Sec. (4.2). In Sec. (5.1) we discuss the $U$-macrovariables and the corresponding Boltzmann entropy $S_{B}^{U}$. In Sec. (5.2) we define the single-particle basis of states that are localized both in position and momentum space. This is then used to define the $f$-macrovariables and the Boltzmann entropy $S_{B}^{f}$ for the quantum gas. In Sec. (6), we present the numerical results. The results for fermions are presented in Sec. (6.1) and for bosons in Sec. (6.2). We discuss and summarize our findings in Sec. (7).

We now discuss the construction of macrostates and the Boltzmann entropy for isolated macroscopic quantum systems. For a more detailed discussion see Refs. goldstein2017 ; tasaki2016 ; mori2018 . Consider a quantum system with a Hilbert space $\mathcal{H}$, whose wave function lies in an energy shell $\mathcal{H}_{E}$ of width $\Delta E\ll E$. As macro observables, one option is to choose a set of commuting coarse-grained operators $\{\hat{M}_{k}\},\leavevmode\nobreak\ k=1,\ldots,J$, meaning that the eigenvalues of each operator are grouped into “bins” and all eigenvalues of that operator within each bin are set equal. The simultaneous diagonalization of all the $\hat{M}_{k}$ operators then provides a decomposition of the accessible Hilbert space into a sum of orthogonal subspaces $\mathcal{H_{\nu}}$,

where $\nu=(\nu_{1},\ldots,\nu_{J})$ defines a macrostate, and $\mathcal{H}_{\nu}$ is the joint eigenspace of the $\hat{M}_{k}$ with eigenvalues $\nu_{k}$. The $\mathcal{H}_{\nu}$ will be referred to as macro-spaces, with $|\mathcal{H}_{\nu}|={\rm dim}\leavevmode\nobreak\ \mathcal{H}_{\nu}$ the dimension of the corresponding macro-space. One then associates a Boltzmann entropy $S_{\nu}=k_{B}\ln|\mathcal{H}_{\nu}|$ to each of the macro-spaces. Let us denote by $\hat{P}_{\nu}$ the projector onto the space $\mathcal{H}_{\nu}$. Any microstate, which is a pure state $|\Phi\rangle$, is said to be in the macrostate $\nu$ if $|\Phi\rangle$ is almost in $\mathcal{H}_{\nu}$, i.e. if $\langle\Phi|\hat{P}_{\nu}|\Phi\rangle\approx 1$, and then its Boltzmann entropy is given by $S_{B}(|\Phi\rangle)=S_{\nu}$. It is expected for many systems, including the one we study here, that for appropriate physical macrovariables if the initial pure state $|\Phi(0)\rangle$ is in a given macrostate, the time-evolved microstate $|\Phi(t)\rangle$ will continue to be (at almost all times) in one single macrostate, i.e. Schrodinger cat-like states will not occur. The exception being when the system crosses from one macrostate to another.

As in the classical case, for a physical choice of the macrovariables, a particular macro-space has by far the largest dimension, which we refer to as the equilibrium macro-space and denote it by $\mathcal{H}_{\rm eq}$. It is characterized by the fact $|\mathcal{H}_{\rm eq}|=(1-\epsilon)|\mathcal{H}_{E}|,\leavevmode\nobreak\ \epsilon\ll 1$, which we take as a physical requirement for any proper choice of the macrovariables. We also say that a system is in equilibrium when its microstate $|\Phi\rangle$ is in, or almost in, $\mathcal{H}_{\rm eq}$.

Now consider the unitary time-evolution of a system that is initially prepared in a non-equilibrium pure quantum state $|\Phi(0)\rangle$. Non-equilibrium means that $|\Phi(0)\rangle$ is not in, or not almost in, the space $\mathcal{H}_{\rm eq}$. It starts in one of the other macro-spaces $\mathcal{H}_{\nu\neq{\rm eq}}$ and with time it moves between different macro-spaces until it eventually ends up in the equilibrium macro-space and stays there for almost all subsequent times. It is expected that the non-equilibrium system should evolve to macro-spaces of higher dimensions, leading to a monotonic growth of entropy. This is what we would like to demonstrate in an explicit example. It is important to note that we are able to define this Boltzmann entropy for the pure quantum state, $|\Phi(t)\rangle$, at any time griffiths1994 .

Computing the Boltzmann entropy: We note that the entropy $S_{\nu}$ of the macrostate $\nu$, and thus $S_{B}(|\Phi(t)\rangle)$ for $|\Phi(t)\rangle$ in ${\cal{H}}_{\nu}$, is equal to the Gibbs-von Neumann entropy of a generalized microcanonical ensemble, which is a mixed state that is uniform over (the unit sphere of) ${\cal{H}}_{\nu}$. It is important to stress, however, that this generalized microcanonical ensemble is not what we take as an accurate microscopic description of the microstate; it is only being used as a construction to compute the Boltzmann entropy of the microstate $|\Phi(t)\rangle$ at time $t$. The Boltzmann entropy at a later time $t^{\prime}>t$ is dictated by what macrostate the unitarily time-evolved microstate $|\Phi(t^{\prime})\rangle$ at that time corresponds to. If we were to instead unitarily time-evolve the density matrix of this out-of-equilibrium generalized microcanonical ensemble, its Gibbs-von Neumann entropy would not change in contrast to the second law of thermodynamics. A typical out-of-equilibrium microstate will, as time advances, typically move to other macrostates of higher Boltzmann entropy, so the Boltzmann entropy thereby does obey the second law for almost all microstates. The macrostate does not time-evolve unitarily, due to it being constructed via a coarse-graining with suitably chosen macrovariables.

This generalized microcanonical ensemble that is used to calculate the Boltzmann entropy of the macrostate at time $t$ may be replaced by an equilibrium ensemble for a fictitious system where constraints have been imposed on all macrovariables to have particular values $\nu$. Then, since that is a macroscopic equilibrium system, the extensive part of its microcanonical entropy should be equal to the extensive part of the Gibbs-von Neumann entropy of an equivalent generalized canonical ensemble. This shows that we can arrive at a correct (to leading order) count of the number of independent microstates in ${\cal{H}}_{\nu}$ by calculating the Gibbs-von Neumann entropy of an equivalent generalized canonical ensemble. This is convenient because in many cases, calculating properties is simpler in the canonical ensemble than in the microcanonical ensemble.

If we accept, as is argued above, that the Boltzmann entropy of the microstate $|\Phi(t)\rangle$ of a macroscopic system is equal, to leading order in system size, to the Gibbs-von Neumann entropy of a properly chosen generalized canonical ensemble, this allows us to skip the step of defining the generalized microcanonical ensemble. We use the expectation values, $\langle\hat{M}_{k}\rangle=\langle\Phi(t)|\hat{M}_{k}|\Phi(t)\rangle,\leavevmode\nobreak\ k=1,\ldots J,$ to define the “equivalent” generalized canonical (GC) ensemble as

and the Lagrange multipliers $\lambda_{k}$ are obtained from the constraint equations

The resulting Boltzmann entropy $S_{B}$ for the microstate $|\Phi(t)\rangle$ and the choice of $\{\hat{M}_{k}\}$ is then given by

We note that this is also equal to the maximum value of the Gibbs-von Neumann entropy

over all $\hat{\rho}_{N}$, subject to the constraints $Tr[\hat{M}_{k}\hat{\rho}_{N}]=\langle\hat{M}_{k}\rangle$ fick1990 .

Using this generalized canonical approach to compute the Boltzmann entropy permits more flexibility in the choice of macrovariables, as compared to those that adhere to the conditions needed for the above construction of generalized microcanonical macrostates. In particular, the spectra of the operators $\{\hat{M}_{k}\}$ need not be coarse-grained, and for quantum systems, these operators need not commute and may be microscopic operators, as we demonstrate in the example below.

In this work, we consider a quantum ideal gas of $N$ particles on a circle of length $L$. We examine the Boltzmann entropy for two different choices of macrovariables, schematically described in Fig. (1), that are quantum analogs of those used for the classical ideal gas in Ref. chakraborti2021entropy :

1. 1.

   $U$-macrovariables: These are specified by the three conserved fields of particle number, momentum, and energy and are direct analogs of the corresponding classical macrovariables chakraborti2021entropy . They correspond to the usual fields that are used in the hydrodynamic description of interacting systems — our non-interacting system has further conserved quantities but we can choose to consider the coarse-grained description in terms of these fields alone. More specifically, we divide our system of length $L$ into $A$ spatial cells of size $\delta=L/A$. We consider a set of operators $\{\hat{N}_{\ell},\hat{P}_{\ell},\hat{E}_{\ell}\},\leavevmode\nobreak\ \ell=1,2\ldots A$, corresponding to the total number of particles, the total momentum, and the total energy in the $\ell^{\text{th}}$ cell. The precise definitions of these operators are given in Sec. (5.1). Given that the system is in a microstate $|\Phi(t)\rangle$, let $\langle\hat{B}\rangle=\langle\Phi|\hat{B}|\Phi\rangle$ denote the expectation value of any operator $\hat{B}$. Then, the set of expectation values $\{N_{\ell},P_{\ell},E_{\ell}\}=\{\langle\hat{N}_{\ell}\rangle,\langle\hat{P}_{\ell}\rangle,\langle\hat{E}_{\ell}\rangle\}$ specify, as we argued in the preceding paragraphs, the $U$-macrostate.

2. 2.

   $f$-macrovariables: For the classical gas, the $f$-macrovariables are the distribution $f(x,p)$ of the single-particle positions and momenta chakraborti2021entropy . Here we do not use a direct quantum analog of the classical macrovariables, but instead use a more microscopic choice, which is permitted for the quantum gas: We choose a complete and orthonormal basis of single-particle wavefunctions, each of which is approximately localized in position space within a width $\Delta x$ and in momentum space with width $\Delta p$. Such states will necessarily have to satisfy the uncertainty relation $\Delta x\Delta p\gtrsim\hbar$. As we will show in Sec. (5.2), it is possible to construct such an orthonormal basis set on the circle with states that are approximately localized in position space with $\Delta x=L/K$ and in momentum space with $\Delta p=2\pi\hbar K/L$, where $K$ is an integer. We denote these single-particle basis states by $|\psi_{\alpha}\rangle$ with $\alpha\equiv(r,v)$, where $r=1,\ldots,K$ and $v=\vartheta K$ where $\vartheta\in\mathbb{Z}$. These states are localized around $x=r\Delta x$ and $p=\vartheta\Delta p$. Let $\hat{n}_{\alpha}$ be the number operator corresponding to the occupation of the state $\ket{\psi_{\alpha}}$. The commuting operators $\{\hat{n}_{\alpha}\}$ form a set of microvariables, since specifying all of their eigenvalues specifies the microstate completely.

   To define macrovariables for the quantum gas, we consider the expectation values $\{D_{\alpha}=\expectationvalue{\hat{n}_{\alpha}}\}$ in the microstate $\ket{\Phi}$. Since a pure state wavefunction of a quantum system implies a probabilistic description, specifying only the $\{D_{\alpha}\}$ implies a coarse-graining, in the sense that they do not specify the microstate. In fact, these expectation values $\{D_{\alpha}\}$ constitute a good set of macrovariables that can be used to define a corresponding generalized canonical ensemble and Boltzmann entropy, following the procedure outlined above. The resulting Boltzmann entropy is

   $\displaystyle\hskip 17.07182ptS_{\text{B}}^{f}=\sum_{\alpha}\left[-D_{\alpha}\ln D_{\alpha}\mp(1\mp D_{\alpha})\ln(1\mp D_{\alpha})\right],$

   (7)

   where the negative and positive signs in the $\mp$ are for fermions and bosons, respectively. This entropy can also be described from a generalized grand canonical ensemble, where each of the single-particle states $\alpha$ has its own chemical potential $\mu_{\alpha}$ in order to set $D_{\alpha}$. For the classical gas, the cells $\Delta x\Delta p$ used to define the $f$-macrovariables must be large enough so that they are typically occupied by many particles, in order to have enough coarse-graining chakraborti2021entropy . For the quantum gas, on the other hand, fully microscopic cells may be used, and the needed coarse-graining is provided by using the expectation values as the macrovariables.

Our system is a quantum ideal gas of $N$ particles on a circle of length $L$. The microstate of the system is a pure state $|\Phi(t)\rangle$ or, using the position representation, the wavefunction $\Phi(x_{1},x_{2},\ldots,x_{N},t)$. This state evolves in time via the Schrödinger equation $i\hbar\partial_{t}\Phi=\hat{H}\Phi$ with the free circle Hamiltonian $\hat{H}=-[\hbar^{2}/(2m)]\sum_{\ell=1}^{N}\partial_{x_{\ell}}^{2}$. The gas is initially confined to a part of the circle, of length $aL$ with $0<a<1$ (usually $a=1/2$), which we refer to as a “box”. The initial $N$-particle state $\ket{\Phi(0)}$ may be taken to be an eigenstate of the occupations of the single-particle energy eigenstates in the box. We also compare our results to the initial grand canonical mixed states in the box. At $t=0$, the box walls are removed at both ends and the gas is allowed to freely expand on the full circle. As the gas expands, we are interested in the time evolution of the macrovariables $U$ and $f$ and of the corresponding entropies.

The single-particle energy eigenstates and eigenvalues on the circle are given by

where $n$ runs over all integers. These are also momentum eigenstates with eigenvalues $p_{n}=2\pi n\hbar/L$. A complete $N$ particle basis of Fock states is specified by the occupations $|\{N_{n}\}\rangle$ with $N_{n}\in\{0,1\}$ for fermions and $N_{n}\in\{0,1,\ldots,\infty\}$ for bosons, constrained so that $\sum_{n}N_{n}=N$. We will find it useful sometimes to use the language of second quantization and so we define the particle creation and annihilation operators $\hat{\Psi}^{\dagger}(x),\hat{\Psi}(x)$ which create or annihilate, respectively, a particle at position $x$. For fermions, they satisfy the anti-commutation relation $\{\hat{\Psi}(x),\hat{\Psi}^{\dagger}(x^{\prime})\}=\delta(x-x^{\prime})$, while for bosons they satisfy the commutation relation $[\hat{\Psi}(x),\hat{\Psi}^{\dagger}(x^{\prime})]=\delta(x-x^{\prime})$. We also define the creation and annihilation operators $\hat{b}^{\dagger}_{n}$, $\hat{b}_{n}$, corresponding to the single-particle energy eigenstate $\varphi_{n}$. These are related to the position operators as $\hat{\Psi}(x)=\sum_{n}\hat{b}_{n}\varphi_{n}(x)$. In the Heisenberg representation, these operators have simple time evolution for our noninteracting gas

As we will see in the subsequent sections, for our purposes, it suffices to study the single-particle density matrix and the corresponding Wigner function. Therefore, in the following subsections, we discuss the evolution of the density matrix and the Wigner function.