Entropy Non-Conservation and Boundary Conditions for Hamiltonian Dynamical Systems

Entropy non-conservation and boundary conditions for Hamiltonian dynamical systems

Gerard McCaul∗ Tulane University, New Orleans, LA 70118, USA and King’s College London, London WC2R 2LS, U.K.

Alexander Pechen† Steklov Mathematical Institute of Russian Academy of Sciences, Gubkina str. 8, Moscow 119991, Russia and National University of Science and Technology "MISIS", Leninski prosp. 4, Moscow 119049, Russia

Denys I. Bondar‡ Tulane University, New Orleans, LA 70118, USA (Dated: June 24, 2019) Applying the theory of self-adjoint extensions of Hermitian operators to Koopman von Neumann classical mechanics, the most general set of probability distributions is found for which entropy is conserved by Hamiltonian evolution. A new dynamical phase associated with such a construction is identified. By choosing distributions not belonging to this class, we produce explicit examples of both free particles and harmonic systems evolving in a bounded phase-space in such a way that entropy is nonconserved. While these nonconserving states are classically forbidden, they may be interpreted as states of a quantum system tunneling through a potential barrier boundary. In this case, the allowed boundary conditions are the only distinction between classical and quantum systems. We show that the boundary conditions for a tunneling quantum system become the criteria for entropy preservation in the classical limit. These findings highlight how boundary effects drastically change the nature of a system.

I. INTRODUCTION that Hamiltonian systems are time reversible. The two ideas are often combined into the claim that systems with Entropy is a critical element of physical theories, defin- a Hamiltonian evolution conserve entropy. This is not in ing the notion of equilibrium [1], as well as serving as a fact the case. The probability density ρ of a Hamiltonian measure of irreversibility [2] and the arrow of time [3–5]. system evolves according to the Liouville equation In thermodynamics, the physical content of entropy is ρ˙ = {H, ρ} . (2) contained in its time dynamics, rather than its absolute value. Given the vital role which entropy plays in char- Taking the time derivative of Eq. (1), substituting the acterizing physical systems, the conditions under which Poisson bracket {·, ·}, and integrating this by parts, one entropy is time dependent are of great interest. finds that the entropy production arises solely from the While entropy is an integral component of our under- boundary terms. For a system with box boundaries at standing of thermodynamics and information, it is not q±, p±, a microscopic quantity and its definition therefore con- " #p+ " #q+ Z q+ ∂2H Z p+ ∂2H tains a degree of flexibility [6]. The entropy of a system S˙ = dqρ − dpρ depends on the level of course graining in the phase space, q− ∂q∂p p− ∂q∂p p− q− and on the variables considered [7]. In this paper we shall " #q+ Z p+ ∂H adopt the Gibbs measure of entropy for a probability den- + dpρ (ln ρ + 1) sity ρ of a one-dimensional system with coordinate q and p− ∂p momentum p, q− " #p+ Z q+ ∂H Z − dqρ (ln ρ + 1) (3) S = − dqdpρ ln ρ (1) q− ∂q p− It is at this point that one generally assumes boundary

arXiv:1904.03473v2 [cond-mat.stat-mech] 21 Jun 2019 which at equilibrium is equivalent to the thermodynamic entropy [8]. conditions to kill these terms [9], but this assumption It is often stated that, for a reversible process, the is overly restrictive, and peculiar to both the dynamics entropy change is zero. Another common assumption is and distribution which is being evolved. This assumption does not give a general answer to the question of entropy conservation. At the same time, it is difficult to assess how modifications, such as a restriction in phase space, ∗ [email protected] would affect the entropy production. † [email protected] To answer the question of entropy conservation with ‡ [email protected] the greatest possible generality, we shall employ the 2

Koopman von-Neumann (KvN) formalism. In particu- the Hilbert space H = L2 (P, dµ): lar, we shall use this formalism to reduce the problem Z of entropy conservation to the analysis of a classical self- 2 2 H = L (P, dµ) = φ : P → C dµ |φ| < ∞ , (4) adjoint evolution operator. The theory of self-adjoint extensions of Hermitian op- which is the set of all functions on the space P that are erators is closely related to boundary conditions, and is square integrable with measure dµ. In classical KvN dy- vital in the resolution of apparent paradoxes in quantum namics P is a phase space, while in quantum mechanics mechanics [10]. The theory also makes explicit connec- it is a configuration space. The inner product of the ele- tion to topological phases of matter [11–13] and spon- ments of this Hilbert space is taneously broken symmetries [14–16]. Moreover, self- adjointness assumes a vital role in novel approaches to Z ∗ prove the Riemann zeta hypothesis [17]. It is, in short, a hφ |ψ i = dµφ ψ, (5) useful formalism that may be applied to the problem of entropy conservation when alloyed to the KvN descrip- for φ, ψ ∈ H. The elements of H are interpreted as (un- tion of classical dynamics. normalized) probability density amplitudes which obey We proceed as follows: A brief overview of KvN me- the Born rule in both quantum and KvN mechanics [37], chanics is provided in Section II. Section III outlines the 2 theory of self-adjoint operators necessary for our analy- |ψ| = ρ. (6) sis, and the conditions under which a Hermitian opera- In both cases, the dynamics of elements of H are given tor is self adjoint. The equivalence of the criteria for a by a one-parameter, strongly continuous group of unitary generator of time translations to be self-adjoint and for transformations on H, that evolution to conserve entropy is shown. Section IV applies the developed technique to a classical oscillator, ψ(t) = U(t)ψ(0). (7) and establishes the conditions for entropy conservation when its coordinate is restricted to the half-line (i.e., a The form of this family of unitary transformations is, by phase space P = R+ × R). Section V provides specific Stones’ theorem [38]: counterexamples of distributions evolved by Hamiltonian dynamics which violate entropy conservation. Section VI U (t) = eiAtˆ , (8) discusses in detail these entropy violating distributions, providing an interpretation of their physical significance where Aˆ is a unique self-adjoint operator. Self- and relation to quantum dynamics. Finally, Sec. VII adjointness is a stronger condition than Hermiticity, and concludes with a discussion of the results. its necessity shall be demonstrated in Sec. III A. The family of unitary transformations may be interpreted as time evolution, leading to the differential equation: II. KOOPMAN-VON NEUMANN DYNAMICS ψ˙ = iAψ.ˆ (9)

KvN mechanics is the reformulation of classical me- Specifying the form of this operator adds physics to the chanics in a Hilbert-space formalism [18]. This opera- formalism. For example, one may derive both quantum tional formalism underlies ergodic theory [19], and is a and classical dynamics from this model by defining a fun- natural method for modeling quantum-classical hybrid damental commutation relation and assuming that ob- systems [20–22]. KvN dynamics has also been applied servable expectations obey the Ehrenfest theorems. to establish a classical speed limit for dynamics[23] and The sole distinction between Koopman dynamics and an alternate formulation of classical electrodynamics [24]. quantum mechanics is in the choice of commutation re- By using the KvN formalism it is even possible to combine lations [25]. In the quantum case [ˆx, pˆ] = i~, and the classical and quantum dynamics in a unified framework self-adjoint operator is then Aˆ = − 1 Hˆ , yielding: known as operational dynamical modeling [25, 26]. ~ ˙ ˆ The KvN formalism has been successfully applied to i~ψqm = Hψqm, (10) construct both deterministic and stochastic classical path integrals [27, 28], including generalizations with geomet- which is the familiar Schrödinger equation. In KvN dy- ric forms [29]. These path integral formulations may be namics, [ˆx, pˆ] = 0, and Aˆ is the Koopman operator Kˆ usefully applied with classical many-body diagrammatic defined as methods [30] and for the derivation and analysis of gen- Kψˆ = −i {H, ψ } . (11) eralized Langevin equations [31]. KvN has been used to cl cl study specific phenomena, including examinations of dis- This leads to the remarkable conclusion that probability sipative behavior [32], linear representations of nonlinear density amplitudes are governed by the Liouville equa- dynamics [33], analysis of the time-dependent harmonic tion, in the same way as probability densities themselves: oscillator [34], and other industrial applications [35, 36]. ˙ Both KvN and quantum dynamics begin by defining ψcl = iKψˆ cl = {H, ψcl} . (12) 3

The choice of commutation relation also defines the A Hermitian (or symmetric) [19] operator is such that measure on the Hilbert space. While dµ = dqdp is D(Aˆ) ⊆ D(Aˆ†) and the action of the operator and its a valid measure in the classical case, only dµ = dq (in adjoint is identical on D(A). Then the coordinate representation) preserves non-negativity for the quantum commutation relation [39]. One may hφ|Aψˆ i = hAφˆ |ψi, ∀φ, ψ ∈ D(Aˆ). (15) still use the full phase space quantum mechanically, but the object under consideration in this case is the Wigner In the special case D(A) = D(A†) the operator is called quasiprobability distribution [40–42]. self-adjoint. The condition D(Aˆ) ⊆ D(Aˆ†) requires that It may appear at first glance that the KvN framework the adjoint of a symmetric operator is defined for any is a formal hammer cracking a classical nut, but this for- element in D(Aˆ), but does not otherwise restrict its do- mulation provides a key advantage; one may exploit the main. Figure 1 illustrates the fact that the domain of a deep theory of operators on Hilbert space. In particular, symmetric operator on a Hilbert space is a subset of the the theory of self-adjoint operators allows the same gen- domain of its adjoint operator. eral statements to be made about entropy conservation as To illustrate the difference between a Hermitian and in the quantum case. Explicitly, if Kˆ is self-adjoint, then self-adjoint operator, we evaluate a Koopman operator the Gibbs entropy will be conserved. To prove this how- with domain ever, we must first review the definition of a self-adjoint ∂ψ ∂ψ operator. D(Kˆ ) = ψ ψ ∈ H such that , ∈ H ∂q ∂p and ψ (q±, p) = ψ (q, p±) = 0 . (16) III. CRITERIA FOR CONSERVATION OF ENTROPY The Hermitian property of this operator may be investi- A. Self-Adjoint Operators gated by using integration by parts: Z ∂H ∂ψ ∂H ∂ψ hφ|Kψˆ i = i dqdp φ∗ − An operator in the Hilbert space H is a pair (D(Aˆ), Aˆ) ∂p ∂q ∂q ∂p ˆ ˆ q p where A is a linear mappings of elements in a subset D(A) Z ∂H + Z ∂H + = i dp φ∗ψ − i dq φ∗ψ of the Hilbert space onto H, ∂p ∂q q− p− Aˆ : D(Aˆ) → H. (13) +hKφˆ |ψi. (17) ˆ The set D(Aˆ) is the domain of the operator, i.e., the set If K is Hermitian, the following boundary condition of vectors in the Hilbert space for which the operator holds: mapping is defined. If the Hilbert space is finite, oper- Z ∂H q+ Z ∂H p+ ators may be represented as finite matrices, and multi- dp φ∗ψ − dq φ∗ψ = 0, (18) ∂p ∂q plication of a vector by a matrix is well defined for all q− p− vectors in a finite-dimensional Hilbert space, in this case which is automatically satisfied by using the example do- D(Aˆ) ≡ H [43]. For infinite-dimensional Hilbert spaces main of Eq. (16). As a result, these boundary conditions however, this is not the case and the operator domain for the wavefunction specify the operator’s domain. does not necessarily coincide with the full Hilbert space. ˆ ˆ For the example domain, the boundary condition is Assuming an operator A is densely defined [i.e., D(A) satisfied regardless of the domain of the adjoint operator, is a dense subspace of H] [19], one may define its ad- ˆ† ˆ ˆ† D(K ) ⊇ D(K). This potential mismatch in the domains joint operator A as follows. A vector φ ∈ H belongs of an operator and its adjoint is deeply problematic, as ˆ† to the domain D(A ) if and only if the linear functional it implies a violation of time-reversal invariance [10]. For ˆ fφ(ψ) := hφ|Aψi is continuous. Then, by the Riesz rep- this reason, the generator of time-translations (and in resentation theorem, there exists a unique z ∈ H such fact all observable operators) must be self-adjoint, as de- that hφ|Aψˆ i = hz|ψi for any ψ ∈ D(Aˆ). Action of the fined above. Given an operator with a defined action, adjoint operator on φ is defined as Aˆ†φ = z. One has it is a nontrivial exercise to find which domains, if any, will make the operator self-adjoint. Fortunately, there is hφ|Aψˆ i = hAˆ†φ|ψi ∀φ ∈ D(Aˆ†), ψ ∈ D(Aˆ). (14) a powerful theorem of functional analysis that may be applied to this problem. It may happen that the adjoint operator does not exist (its domain may not be dense or even empty; see, e.g., Example 4 in Sec. VIII.1, Vol. 1 of Ref. [19]). This B. The von Neumann deficiency index theorem occurs if the graph of the operator; that is, the set of pairs (ψ, Aψˆ ), ψ ∈ D(Aˆ), is not closable in H × H. Otherwise The von Neumann deficiency index theorem [44] can the adjoint operator is correctly defined. be used to answer the question of whether an operator 4

Aˆ may be made self adjoint. Details of this important theorem may be found in Refs. [10, 43, 45], but we shall brieﬂy outline its use here. The deﬁciency index theorem determines all possibilities for a Hermitian operator Aˆ with domain D(Aˆ) by considering the eigenstates in D(Aˆ†) with imaginary † eigenvalues, i.e., those ψ± ∈ D(Aˆ ) satisfying the equation:

† Aˆ ψ± = ±iψ±. (19) Figure 1. The domain of a symmetric operator Aˆ is a subset The number of linearly independent solutions for ψ± are of the domain of its adjoint operator. A self-adjoint extension the deﬁciency indices n±. They determine the following modiﬁes the domains of both operators such that they become three possibilities [19]: identical.

n+ = n− = 0 Aˆ is essentially self-adjoint. where n+ = n− = n ≥ 1 Aˆ has infinite self-adjoint extensions. ρˆcl = |ψcl ih ψcl| . (22) n+ 6= n− Aˆ has no self-adjoint extension. Note that the Sˆ operator is explicitly nonlinear; how- A self-adjoint extension is an operator Aˆλ with the ˆ ˆ ever, the proof of its time independence does not rely on same action as A on D(A), but whose domain has been linearity, as if Kˆ is self-adjoint, then modified, D(Aˆ) → D(Aˆλ) ⊇ D(Aˆ), to enforce the self- h † i adjoint condition. Figure 1 shows an example schematic, Tr[Sˆ (t)] = Tr e−iKtˆ Sˆ (0) eiKˆ t = Tr[Sˆ (0)], (23) demonstrating the domain modification made by a self- adjoint extension. where in the final equality, we have exploited that Kˆ is If n+ = n− = n, each self adjoint extension is charac- self-adjoint and all the traces are finite. Having demon- terized by n2 parameters, in the form of an n × n uni- strated the time independence of the trace of Sˆ, we now tary matrix U. For the Koopman operator, the bound- express it in the q, p basis: ary condition corresponding to a self-adjoint extension Z Z ˆ 0 0 0 0 0 0 with domain D(Aλ) is simply Eq. (18), replacing φ with Sˆ = − dqdp ρ |q, p ih q, p| ln ρ dq dp |q , p ih q , p | . n P m ψ + Un,mψ [14]: + m − (24) Expanding this yields: " !∗ #q+ Z X ∂H Z ∀n : dp ψn + U ψm ψ Sˆ = − dqdp ρ ln (ρ) |q, p ih q, p| , (25) + n,m − ∂p m q− " !∗ #p+ and therefore Z X ∂H = dq ψn + U ψm ψ . (20) Z + n,m − ∂q Tr[Sˆ] = − dqdp ρ ln ρ = S. (26) m p−

n ˆ Here ψ± is one of the n solutions to Eq. (19) and Un,m is Thus, we establish that Tr[S] is the entropy, and is con- an element of the unitary matrix U. Consequently, one served for self-adjoint Kˆ . Note, however, that entropy may use the deficiency index theorem to characterize the conservation is not in itself a guarantee of unitary evolu- most general boundary condition on ψ for which Kˆ is tions. self-adjoint in terms of U. To summarize, entropy conservation is guaranteed when the Koopman operator Kˆ is self-adjoint. The domain of the self-adjoint extensions may be determined by C. Self-Adjoint Operators and Entropy using the deficiency index theorem. This domain corre- Conservation sponds to the most general boundary conditions which conserve entropy for a given Hamiltonian. While the von Neumann deficiency index theorem es- Establishing the self-adjoint extensions of an operator tablishes the conditions under which Kˆ is self-adjoint, is a nontrivial exercise, but the analysis is most straight- this does not explicitly address the question of en- forward in a system whose Koopman operator is func- tropy conservation. We now demonstrate that a self- tionally dependent on only one phase-space coordinate. adjoint Koopman operator guarantees entropy conserva- Therefore, in the following we consider the specific ex- tion. First, define the operator [46]: amples of free and periodic systems in action-angle coordinates, but emphasize that this is a choice of computa- Sˆ = −ρˆcl lnρ ˆcl, (21) tional convenience in applying generic arguments. 5

Solutions to this equation have the form

∓ θ ψ±(θ, J) = f±(J)e ω . (32) The solutions which belong to L2 are all those for which:

Z θ+ Z Jb Z Jb Z θ+ 2 2 ∓2 θ dθ dJ |ψ±| = dJ |f± (J)| dθ e ω θ− 0 0 θ− ! Z Jb θ θ ω 2 ∓2 + ∓2 − = ∓ dJ |f± (J)| e ω − e ω < ∞. 2 0 Figure 2. Example phase-space boundaries in (left) original (33) coordinate system and (right) action-angle coordinates. R Jb 2 Any function f± satisfying 0 dJ |f± (J)| < ∞ determines a particular solution to ψ±. From this we conclude IV. ENTROPY CONSERVATION FOR CYCLIC that n = n = ∞. SYSTEMS + − Rather than apply Eq. (20) directly, in this case the Koopman operator has the same form as the quantum- We now apply the results of the previous section to de- mechanical momentum operator on a fixed interval. Fol- rive the most general entropy-preserving boundary con- lowing the examples of Refs. [10, 14] we conclude that ditions for both the harmonic oscillator and free particle for each value of J, the Koopman operator has a one with a bounded phase space. The first section will con- parameter self-adjoint extension. Labeling each of these sider general box boundaries in action-angle coordinates. arbitrary parameters by β(J), we find the boundary condition for the self-adjoint domain of Kˆ :

A. Self-adjoint domain In action-angle coordinates iβ(J) ψ (θ+,J) = e ψ (θ−,J) . (34)

Consider a system whose coordinates can be canoni- While this phase is not directly observable in expecta- cally transformed into a space where one coordinate is tions, it is gauge-invariant (in the sense of locally rotat- cyclic, i.e., a Hamiltonian that functionally depends on ing a complete set of states). only one phase-space coordinate. Any periodic system Potential physical consequences of the choice of self- may be described by these canonical action-angle coor- adjoint extension can be examined by observing that time dinates [47], but here we take the simplest example of a evolution corresponds to a translation in θ. Including the harmonic oscillator: time argument explicitly in the wavefunction, harmonic dynamics guarantee ω2q2 + p2 H = . (27) 2 ψ (θ, J, t) = ψ (θ − ωt, J, 0) . (35) Using action-angle coordinates (θ, J) greatly simplifies In particular, the wavefunction at the θ boundaries may the analysis, and can be achieved with the substitution: be expressed in terms of a time translation of the wave- r 2J √ function at an arbitrary point in θ q = sin θ, p = 2ωJ cos θ. (28) ω θ − θ ψ (θ , J, t) = ψ θ, J, t + ± . (36) In the action-angle representation, we choose box phase- ± ω space boundaries, with θ ∈ [θ−, θ+] and J ∈ [0,Jb] (see Fig. 2), which contains a particle restricted to the half- Combining this with the fact that Eq. (34) must hold at line q > 0 as a special case. In the new coordinates, all times, we obtain the relation H = ωJ, leading to the following form of the Koopman Nθ operator for the harmonic oscillator: ψ θ, J, t + b = eiNβ(J)ψ (θ, J, t) , (37) ω ∂ψ Kψˆ = −i {H, ψ} = −iω . (29) ∂θ where N is an integer and θb = θ+ − θ−. When θb is 2π chosen to be N , states may acquire an additional phase We now apply the deficiency index theorem in this new iNβ(J) basis by finding the wavefunctions satisfying e after each period of the motion. For this reason, although the dynamics are periodic, it is possible Kψˆ ± = ±iψ±, (30) to choose boundary conditions such that the wavefunction is not periodically symmetric due to the additional meaning that we look for solutions to phase. While observable expectations will be unaffected ∂ψ 1 by such a phase, the time translation symmetry of corre- ± = ∓ ψ . (31) ∂θ ω ± lation functions will be altered by its presence. 6

To demonstrate this, consider two observables A and which automatically satisfies Eq. (18). B which have the two-time correlation function: This condition can be given a physical interpretation Z by considering the Liouville evolution as a continuity C(t, t0) = dθdJ ψ∗ (t) ABψ(t0), (38) equation, where j is the probability current in phase space: where phase-space arguments on the right-hand side have ∂ρ been omitted for brevity. If one of the time arguments is = −∇ · j, (44) shifted by the period of motion T = 2π , the correlation ∂t ω ∂H ! function becomes jq ∂p j = = ∂H ρ. (45) Z jp − C(t + T, t0) = dθdJ e−iNβ(J)ψ∗ (t) ABψ(t0) ∂q The probability current in the q direction (integrated over 6= C(t, t0). (39) p) Jq [ρ (q, p)] is given by Thus, the phase factor determining a self-adjoint exten- Z Z sion has a nontrivial effect on time symmetries of ob- Jq [ρ (q, p)] = dp jq (q, p) = dp pρ (q, p) . (46) servable correlations. This breaking of periodicity is in sharp contrast with correlation functions on the full Substituting Eq. (43) into the above equation, we find phase space, where continuity of the wave function forces that at the q = 0 boundary, β(J) = 0 and hence C(t, t0) = C(t + T, t0). Jq [ρ (0, p)] = 0. (47) The existence of a dynamical phase due to the boundary conditions is somewhat analogous to the Berry phase From this boundary condition, we conclude that the do- [48] and its classical equivalent the Hannay angle [49]. main of states for self-adjoint Koopman operators cor- The origin of these geometric phases is quite different, re- responds precisely to a reflecting boundary condition at sulting from adiabatic holonomic variation of the Hamil- the q = 0 boundary [51]. tonian parameters. In both cases however, the naive ex- Finally, we note that there is no ω dependence in Eqs. pectation that the system will return to its original state (42) and (47). In the limit ω → 0, the boundary con- (after either a single period of the motion, or returning to dition for entropy preserving distributions is therefore the original Hamiltonian parameters) is not always true. unaffected, but the system Hamiltonian now describes It is well known that, for the simple harmonic oscillator, a free particle rather than a harmonic oscillator. For this the geometric phase change is zero [50], whereas here a reason, Eq. (47) also describes the self-adjoint domain phase may be acquired by the choice of boundary condi- of the free particle. tions.

V. NONCONSERVING STATES B. Entropy conservation on the half-line Given the self-adjoint conditions derived in the previ- Angular restrictions in the action-angle coordinate sys- ous section, one can construct a reasonable initial wave- tem may correspond to unphysical restrictions in the function (and hence probability density) on the half line momentum subspace of the phase space. A restriction which does not conserve entropy. Take for example the purely in the coordinate of phase space may be obtained initial wavefunction by choosing the boundaries − 1 − 1 (p−p )2− 1 q2 ψ (q, p) = Z 2 e 2 0 2 , (48)

Jb = +∞, θ− = 0, θ+ = π, (40) where crucially, p0 6= 0 and the normalization factor is which leads to the half-line q ≥ 0 phase space in the Z = π/2. The rate of entropy change for this state can be calculated directly from Eq. (3). For the free particle original coordinates. The phase space is then P = R+×R and the Hilbert space is H = L2(P, dqdp). Converting we have in the domain q ≥ 0 the wavefunction back to its original coordinates at the √ 1 S˙ = −p Z−1 π − ln Z boundary we obtain FP 0 2 √ ψ (θ ,J) → ψ (0, ±p) (41) 2 2 e ± = −p √ ln ≈ −0.05p 6= 0, 0 π π 0 which may be substituted into Eq. (34) to give the domain of self-adjoint evolutions on the half-line: and for the harmonic oscillator in the same domain (set- ting ω = 1) ψ(0, −p) = eiβ(p)ψ(0, p). (42) √ 5 S˙ = πp Z−1 ln Z + + 4p2 Entropy preserving probability distributions therefore HO 0 2 0 obey the condition 2p π 5 = √ 0 ln + + 4p2 . ρ (0, −p) = ρ(0, p). (43) π 2 2 0 7

In both cases the entropy is nonconserved. The energy expectation and partition function for both systems are also time dependent. For this state the integrated boundary probability current is √ Jq [ρ (0, p)] = −2 πp0, (49) i.e., at the boundary the probability is being either par- tially absorbed or generated depending on the sign of p0.

VI. INTERPRETING NONCONSERVING STATES Figure 3. Schematic of allowed states with an additional potential barrier U0Θ(−q) in both the quantum and classical regimes. The space is partitioned into regions I and II. ρI (q, p) To give an interpretation to the nonconserving states, is a probability distribution satisfying Eq. (47), the reflecting we now consider the boundary conditions for a quantum- boundary condition, and is therefore confined to region I. The mechanical system in the full phase space R2. The same dynamics also describe a quantum Wigner distribution dynamics are governed by the Hamiltonian H (q, p) = W (q, p). In the quantum case however, more general bound- H0 (q, p) + U0Θ(−q), where H0 is the Hamiltonian for ei- ary conditions are given by Eqs. (61) and 62), which allow for ther a free-particle or harmonic oscillator, Θ is the Heav- tunneling, and nonzero Wigner distributions in the classically forbidden region II. iside step function, and U0 is a constant that satisfies

sup [H0 (q, p)] < U0, (50) ρ(q,p)>0 When the system Hamiltonian is at most quadratic in both the p and q coordinates (as is the case for the exam- i.e., the U0Θ(−q) term represents a classically impene- ple Hamiltonians studied in Sec. IV), the Moyal bracket trable barrier on the negative part of the real line. is simplified, and the Wigner distribution is evolved by We describe the quantum system with the wavefunc- the Poisson bracket [52], in the same way as Eq. (2): tion ψ . This can be brought into contact with earlier Q ∂W results by using the Wigner distribution W (x, p) to de- = {H,W } . (55) scribe the quantum system in phase space [52]: ∂t Remarkably, in this case, the quantum Wigner dis- Z tribution, classical probability distribution, and classical 1 − i py y ∗ y W (q, p) = dy e ~ ψQ q + ψQ q − wave function all share the same equation of motion. In 2π~ 2 2 (51) this scenario, quantum and classical systems are distin- The Wigner distribution is a quasiprobability that cap- guished purely by the restrictions placed on them by their tures quantum-mechanical expectations as a distribution boundary conditions. over phase space, which in the classical limit becomes the Labeling the positive half-line as region I and the neg- classical wavefunction [53]. Here, the classical wavefunc- ative half-line as region II (see Fig. 3), we consider a tion refers to the wavefunction evolved according to KvN system initially confined to region I. By Eq. (50), it is dynamics, i.e., ψ from Secs. II-V. impossible for a classical system to transition into region For an arbitrary quantum operator Aˆ its expectation II. In this case the classical system is described by the may be described by using the Wigner distribution: phase space P = R+ × R and the results of section IV B apply. Most importantly, Eq. (47) holds, meaning that Z at all times ρII (q, p) = 0 and J ρI (0, p) = 0. hAî = dqdp W (q, p) A (q, p) , (52) q For the quantum system, while the equation of motion for W (q, p) is identical to that for both the classical where A (q, p) is the Wigner map of the operator Aˆ: wave function ψ and the associated probability density ρ, Z a quantum state initially confined to region I may tunnel 1 − i py D y y E ~ ˆ into the classically forbidden region II. For such a quan- A (q, p) = dy e q + A q − . (53) 2π~ 2 2 tum system, the boundary condition between regions I The Wigner distribution evolution is described by the and II is not described by Eq. (50), but must be general- Moyal bracket [53]: ized. This quantum boundary condition simply enforces continuity for the quantum wavefunction ψQ(q) on the ∂W = {{H,W }} border between regions I and II [54]: ∂t I II ←−−→ ←−−→! ψQ(0) = ψQ (0), (56) 2 ~ ∂ ∂ ~ ∂ ∂ = H (q, p) sin − W (q, p) . ∂ψI (q) ∂ψII (q) ~ 2 ∂q ∂p 2 ∂p ∂q Q = Q . (57) ∂q ∂q (54) q=0 q=0 8

To account for this boundary condition, the Wigner dis- above a barrier. This is another purely quantum effect, tribution over the whole real-line is expressed piecewise which may be regarded as tunneling through a momentum space barrier [57]. Performing an analogous classical W I (q, p) for q ≥ 0 W (q, p) = . (58) analysis for a harmonic system restricted to the half-line W II (q, p) for q < 0 in momentum space, a similar relationship between allowed states and boundary conditions can be obtained, The quantum boundary conditions can then be expressed where classically forbidden reflected Wigner distributions in terms of the Wigner function with [52] match entropy nonconserving states of the classical sys- Z ∞ tem. 1 i qp q ψQ(q) = ∗ dp e ~ W ( , p) (59) ψ (0) −∞ 2 and VII. CONCLUSIONS ∂ψ (q) i Z ∞ Q = dp pW (0, p) ∗ We have applied the Koopman von Neumann formal- ∂q q=0 ~ψ (0) −∞ Z ∞ ism to reduce the problem of entropy conservation in a 1 ∂ classical system to the problem of identifying self-adjoint + ∗ dp W (q, p). (60) ψ (0) ∂q q=0 −∞ extensions of the Koopman operator. In this way, one can explore for a given system the full range of admis- Substituting these expressions into the quantum bound- sible, physical probability distributions and phase-space ary conditions of Eqs. (56) and (57), yields these bound- restrictions which preserve entropy. Applying this tech- ary conditions in terms of Wigner distributions: nique to the harmonic oscillator and free particle, a relationship between the choice of self-adjoint extension and Z ∞ Z ∞ the boundary condition was determined. In the case of dp W I (0, p) = dp W II (0, p) , (61) the harmonic oscillator, this choice is reflected in the −∞ −∞ breaking of the periodic symmetry of correlation func- I II Jq W (0, p) − W (0, p) = tions. This demonstrates a new class of situations where Z ∞ self-adjoint extensions of operators play an important ∂ II I −i~ dp W (q, p) − W (q, p) , role in distinguishing subtle phenomena. ∂q q=0 −∞ (62) In the example cases studied, it was possible to construct apparently reasonable states in the phase space where the left-hand side of the second boundary condi- P = R+ × R which do not belong to the domain of a tion is the Wigner flow [55, 56] over the boundary, de- self-adjoint Koopman operator, and consequently do not fined in the same way as Eq. (47). Focusing exclusively preserve entropy. These states were interpreted using the on region I, the current boundary condition is: fact that the restricted phase space for the classical system corresponds to a classically impenetrable potential Z ∞ I ∂ I II barrier, but a quantum system may tunnel into this for- Jq W (0, p) = i~ dp W (q, p) + F W . bidden region. ∂q q=0 −∞ (63) For both the free particle and harmonic oscillator, Here F W II is some functional dependent only on the quantum and classical dynamical equations coincide, and Wigner distribution of region II. This boundary condi- the two regimes are distinguished purely by the allowed tion is the only feature that distinguishes W I from the boundary conditions. This allowed for the interpretation classical system described by ρ = ρI . Furthermore, in the of entropy nonconserving classical states as partial de- classical limit, ~ → 0 and (for a system initially confined scriptions of tunneling quantum states. Additionally, this to region I), W II = 0. In this case Eq. (63) is equivalent approach demonstrated that the classical boundary con- to Eq. (47), recovering the classical boundary condition. dition corresponding to a self-adjoint Koopman operator Equation (63) also allows one to interpret the entropy is the classical limit of the quantum boundary condition nonconserving states in Sec. V. These states are char- for a tunneling state. acterized by Jq [ρ (0, p)] 6= 0. While this violates the These results highlight the importance of boundary classical boundary condition, if ρ is interpreted instead conditions for fundamental aspects of Hamiltonian evo- as W I , then the non-zero integrated current is consistent lution. While Hamilton’s equations of motion provide a with Eq.(63). Hence, for the free-particle and harmonic local description of the dynamics, the entropy provides oscillator, the entropy nonconserving states in a classical a global characterization of the evolution, and is there- system with a classically impassible potential barrier are fore sensitive to the self-adjointness of the Koopman op- partial solutions for a quantum system tunneling through erator. Even in the case that equations of motion are the barrier. formally time-reversal symmetric, the choice of bound- Finally, we note that the arguments presented in this ary condition can break entropy preservation, and thus section may be applied to the phenomenon of reflection time-reversal symmetry. 9

ACKNOWLEDGMENTS Cambier for suggesting a research direction that has directly led to this work. G.M. and D.I.B. are supported by Air Force Oﬃce of Scientiﬁc Research Young Inves- tigator Research Program (FA9550-16-1-0254). A.P. is The authors would like to thank the referees for their supported by project No. 1.669.2016/1.4 of the Ministry valuable comments, which have inspired a number of the of Science and Higher Education of the Russian Federa- results presented here. D.I.B. is grateful to Jean-Luc tion.

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