In the domain of nondissipative unitary Hamiltonian dynamics, the well-known Mandelstam–Tamm–Messiah time–energy uncertainty relation τ F Δ H ≥ ℏ / 2 provides a general lower bound to the characteristic time τ F = Δ F / | d ⟨ F ⟩ / d t | with which the mean value of a generic quantum observable F can change with respect to the width Δ F of its uncertainty distribution (square root of F fluctuations). A useful practical consequence is that in unitary dynamics the states with longer lifetimes are those with smaller energy uncertainty Δ H (square root of energy fluctuations). Here we show that when unitary evolution is complemented with a steepest-entropy-ascent model of dissipation, the resulting nonlinear master equation entails that these lower bounds get modified and depend also on the entropy uncertainty Δ S (square root of entropy fluctuations). For example, we obtain the time–energy-and–time–entropy uncertainty relation ( 2 τ F Δ H / ℏ ) 2 + ( τ F Δ S / k B τ ) 2 ≥ 1 where τ is a characteristic dissipation time functional that for each given state defines the strength of the nonunitary, steepest-entropy-ascent part of the assumed master equation. For purely dissipative dynamics this reduces to the time–entropy uncertainty relation τ F Δ S ≥ k B τ , meaning that the nonequilibrium dissipative states with longer lifetime are those with smaller entropy uncertainty Δ S .
In the domain of nondissipative unitary Hamiltonian dynamics, the well-known Mandelstam–Tamm–Messiah time–energy uncertainty relation τ F Δ H ≥ ℏ / 2 provides a general lower bound to the characteristic time τ F = Δ F / | d ⟨ F ⟩ / d t | with which the mean value of a generic quantum observable F can change with respect to the width Δ F of its uncertainty distribution (square root of F fluctuations). A useful practical consequence is that in unitary dynamics the states with longer lifetimes are those with smaller energy uncertainty Δ H (square root of energy fluctuations). Here we show that when unitary evolution is complemented with a steepest-entropy-ascent model of dissipation, the resulting nonlinear master equation entails that these lower bounds get modified and depend also on the entropy uncertainty Δ S (square root of entropy fluctuations). For example, we obtain the time–energy-and–time–entropy uncertainty relation ( 2 τ F Δ H / ℏ ) 2 + ( τ F Δ S / k B τ ) 2 ≥ 1 where τ is a characteristic dissipation time functional that for each given state defines the strength of the nonunitary, steepest-entropy-ascent part of the assumed master equation. For purely dissipative dynamics this reduces to the time–entropy uncertainty relation τ F Δ S ≥ k B τ , meaning that the nonequilibrium dissipative states with longer lifetime are those with smaller entropy uncertainty Δ S .
We derive exact relations and general inequalities that extend the usual time-energy uncertainty relations from the domain of unitary Hamiltonian dynamics to that of dissipative dynamics as described by a broad class of linear and nonlinear evolution equations for the density operator. For non-dissipative dynamics, by using the Schroedinger inequality instead of the Heisenberg-Robertson inequality, we obtain a general exact time-energy uncertainty relation which is sharper than the usual Mandelstam-Tamm-Messiah relation τFΔH≥ℏ/2. For simultaneous unitary/dissipative dynamics, the usual time-energy uncertainty relation is replaced by a less restrictive relation that depends on the characteristic time of dissipation, τ, and the uncertainty associated with the generalized nonequilibrium Massieu-function operator which defines the structure of the dissipative part of the assumed class of evolution equations. Within the steepest-entropy-ascent dissipative quantum dynamics of an isolated system introduced earlier by this author, we obtain the interesting time-energy and time-entropy uncertainty relation (2τFΔH/ℏ)2+(τFΔS/kBτ)2≥1. We illustrate this result and various other inequalities by means of numerical simulations.
In the framework of the recent quest for well-behaved nonlinear extensions of the traditional Schrödinger-von Neumann unitary dynamics that could provide fundamental explanations of recent experimental evidence of loss of quantum coherence at the microscopic level, a recent paper [Gheorghiu-Svirschevski 2001 Phys. Rev. A 63 054102] reproposes the nonlinear equation of motion proposed by the present author [see Beretta G P 1987 Found. Phys. 17 365 and references therein] for quantum (thermo)dynamics of a single isolated indivisible constituent system, such as a single particle, qubit, qudit, spin or atomic system, or a Bose-Einstein or Fermi-Dirac field. As already proved, such nonlinear dynamics entails a fundamental unifying microscopic proof and extension of Onsager's reciprocity and Callen's fluctuation-dissipation relations to all nonequilibrium states, close and far from thermodynamic equilibrium. In this paper we propose a brief but self-contained review of the main results already proved, including the explicit geometrical construction of the equation of motion from the steepest-entropy-ascent ansatz and its exact mathematical and conceptual equivalence with the maximal-entropy-generation variational-principle formulation presented in Gheorghiu-Svirschevski S 2001 Phys. Rev. A 63 022105. Moreover, we show how it can be extended to the case of a composite system to obtain the general form of the equation of motion, consistent with the demanding requirements of strong separability and of compatibility with general thermodynamics principles. The irreversible term in the equation of motion describes the spontaneous attraction of the state operator in the direction of steepest entropy ascent, thus implementing the maximum entropy production principle in quantum theory. The time rate at which the path of steepest entropy ascent is followed has so far been left unspecified. As a step towards the identification of such rate, here we propose a possible, well-behaved and intriguing, general closure of the dynamics, compatible with the nontrivial requirements of strong separability. Based on the time–energy Heisenberg uncertainty relation, we derive a lower bound to the internal-relaxation-time functionals that determine the rate of entropy generation. This bound entails an upper bound to the rate of entropy generation. By this extreme maximal-entropy-generation-rate ansatz, each indivisible subsystem follows the direction of steepest locally perceived entropy ascent at the highest rate compatible with the time– energy uncertainty principle.
Abstract We consider the temporal evolution of an isolated system of N particles from a non-equilibrium state of entropy S = S′ to the equilibrium state of maximum entropy, S = S max, S′ ≤ S max. The application of usual density matrix theory to the temporal development of S leads us to dS/dt = 0: The entropy S does not change in time t. Thereby it is irrelevant, whether we consider a non-equilibrium state or an equilibrium state. Consequently, the system cannot irreversibly change by entropy production dS/dt > 0 from S′ to S max. This is a paradoxial result, which contradicts the experience. It can be traced back to the von Neumann equation, which in principle describes reversible processes and hence is unsuitable for calculating the irreversible evolution of the entropy S in time t. Each irreversible process is accompanied by a positive entropy production P = dS/dt ≥ 0 inside the system, which only vanishes in case of the equilibrium state. In order to overcome the above mentioned difficulties, we assign an operator P to the entropy production P, which is defined by the eigenvalue equation P |u〉 = P|u〉 with the state vector |u〉 of the N-particle system. There was an extensive discussion about the relation between the production of entropy P on one hand and the progression of time t on the other. Making use of this concept, we combine the operator of entropy production P with the time-development operator U (t, t 0) of the system and finally deduce the infinitesimal unitary operator U (t+τ,t) = 1 + (i/k)τ P by means of very general assumptions. Here P means the generator of the infinitesimal progression of time τ = t − t 0 and k the Boltzmann constant, representing the atomic entropy unit. Similarly to P we also treat the time t as an observable, defined by t |u〉 = t|u〉. We apply the infinitesimal time-evolution operator U (t+τ,t) to the operator of time t and finally obtain the P – t commutation relation i[ P , t ] = k, which is independent of τ. It shows that the operators P and t do not commute, and hence P and t are not sharply defined simultaneously. Instead we have uncertainties ΔP and Δt on measuring P and t, which are given by the P–t uncertainty relation ΔPΔt ≥ k/2. It readily allows a discussion of the evolution of the entropy S of the isolated system in time t from S′ to S max. Now, the irreversible steps are correctly described by the entropy production P = dS/dt > 0, while the thermal equilibrium is given by P = 0, ΔP = 0, and thus the lifetime of the equilibrium state Δt = ∞. According to the P–t uncertainty relation, the Boltzmann constant k is similarly important to the quantum thermodynamics of irreversible processes like Planck′s constant h to usual quantum mechanics.
In this paper, we outline the main features of the nonlinear quantum evolution equation proposed by the present author. Such an equation may be used as a model of reduced subsystem dynamics to complement various historical and contemporary efforts to extend linear Markovian theories of dissipative phenomena and relaxation based on master equations, Lindblad and Langevin equations, to the nonlinear and far nonequilibrium domain. It may also be used as the fundamental dynamical principle in theories that attempt to unite mechanics and thermodynamics, such as the Hatsopoulos–Gyftopoulos unified theory which motivated the original development of this well-behaved general nonlinear equation for the evolution of the density operator capable of generating irreversible deterministic relaxation to thermodynamic equilibrium from any far nonequilibrium state even for an isolated system.
We consider the temporal evolution of an isolated system of particles from a non-equilibrium state of entropy S = S′ to the equilibrium state of maximum entropy S = S max, S′ ≤ S max. The application of usual density matrix theory to the temporal development of S leads us to the unexpected result P = dS/dt = 0. The entropy does not change in time. This result is valid in case of the equilibrium state. However it is wrong for a non-equilibrium state, which could not irreversibly change from S′ to S max by entropy production P > 0. This contradicts the second law of thermodynamics. The paradoxial result can be traced back to the Liouville theorem. To overcome this difficulty, we introduce the t–P space. The time t and the entropy production P are observables in quantum thermodynamics. They are defined by the eigenvalue equations t|t〉 = t|t〉 and P|P〉 = P|P〉. The operators t and P do not commute. We have a non-vanishing commutator, [t,P] = ik, and consequently a t–P uncertainty relation, ΔtΔP ≥ k/2. The Boltzmann constant k is characteristic of quantum thermodynamics and represents the atomic entropy unit. — The time evolution operator U(τ) = exp[−(i/k)τP] shifts the time t by τ, U(τ)|t〉 = |t + τ〉. The generator of τ is the operator P of entropy production. Then the eigenvector |P〉 is also an eigenket of U(t), and it follows with progressing time τ ≥ 0 that τP ≥ 0 and hence P = dS/dt ≥ 0. This is the second law of thermodynamics, which is correctly reproduced in quantum thermodynamics. — Exchanging −τ for +τ replaces the equations of forward time +τ by the equations of backward time −τ. Independent of the direction of the arrow of time, the entropy of the system always increases or remains constant in the limit of the equilibrium state, dS ≥ 0. This is characteristic of irreversibility, for the system does not return to its initial state on time reversal. — The t–P uncertainty relation allows to discuss the temporal evolution of the expectation value 〈P〉 of entropy production and the lifetime Δt of the transient states, the system passes through on approaching the equilibrium state. In case of a non-equilibrium state 〈P〉, ΔP, and Δt are finite. On approaching the equilibrium state, 〈P〉 and ΔP decrease, consequently Δt increases. Finally, in the case of equilibrium (eq), we have 〈P〉eq = 0, ΔP eq = 0, and Δt eq → ∞. The lifetime of the equilibrium state is unlimited, because the entropy production 〈P〉 and the uncertainty of entropy production ΔP exactly vanish. The temporal evolution of the system can be described by means of a phenomenological equation.
The time–energy uncertainty relation in nonrelativistic quantum mechanics has been intensely debated with regard to its formal derivation, validity, and physical meaning. Here, we analyze two formal relations proposed by Mandelstam and Tamm and by Margolus and Levitin and evaluate their validity using a minimal quantum toy model composed of a single qubit inside an external magnetic field. We show that the ℓ1 norm of energy coherence C is invariant with respect to the unitary evolution of the quantum state. Thus, the ℓ1 norm of energy coherence C of an initial quantum state is useful for the classification of the ability of quantum observables to change in time or the ability of the quantum state to evolve into an orthogonal state. In the single-qubit toy model, for quantum states with the submaximal ℓ1 norm of energy coherence, C<1, the Mandelstam–Tamm and Margolus–Levitin relations generate instances of infinite “time uncertainty” that is devoid of physical meaning. Only for quantum states with the maximal ℓ1 norm of energy coherence, C=1, the Mandelstam–Tamm and Margolus–Levitin relations avoid infinite “time uncertainty”, but they both reduce to a strict equality that expresses the Einstein–Planck relation between energy and frequency. The presented results elucidate the fact that the time in the Schrödinger equation is a scalar variable that commutes with the quantum Hamiltonian and is not subject to statistical variance.