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Herbert W. Zimmermann
Quantum Thermodynamics of Entropy Production and Time – An Approach to Irreversibility
Keywords: Operators of Entropy Production P, Time t, Evolution of Time U(τ), t–P Commutator, t–P Uncertainty Relation, Temporal Evolution of Entropy Production, Second Law of Thermodynamics
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 = Smax, S′ ≤ Smax. 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 Smax 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, ΔPeq = 0, and Δteq → ∞. 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.
Zeitschrift für Physikalische Chemie, Oldenbourg Wissenschaftsverlag
Print ISSN: 0942-9352
Volume: 220, 09/2006
Pages: 1131 - 1158