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International Journal of Quantum Chemistry

On the removal of prigogine's quantum statistical “degeneracy” in the theory of molecular systems

C. A. Chatzidimitriou‐Dreismann, E. Lippert

Published Jun 19, 2009• 1 citation
Publisher

Abstract

In many problems of molecular physics of condensed matter the coupling V of the system of interest with its environment is strong. During a time interval of the order of the characteristic relaxation time ΓTy due to V, both the energy spectrum and the time evolution of the system cannot be determined exactly. A new representation is introduced in which the coupling V does not affect the energy spectrum, but it determines the time evolution. This causes the appearance of a new quantity ΓT, called “intrinsic time.” Some results of this procedure are (a) invariance of the basic physical equations under the new trnsformation, (b) noncommutation of ΓT with the Liouvillian, and (c) existence of a nonvanishing “entropy production” with respect to the intrinsic time.

DOI
10.1002/qua.560220820

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In many problems of molecular physics of condensed matter the coupling V of the system of interest with its environment is strong. During a time interval of the order of the characteristic relaxation time ΓTy due to V, both the energy spectrum and the time evolution of the system cannot be determined exactly. A new representation is introduced in which the coupling V does not affect the energy spectrum, but it determines the time evolution. This causes the appearance of a new quantity ΓT, called “intrinsic time.” Some results of this procedure are (a) invariance of the basic physical equations under the new trnsformation, (b) noncommutation of ΓT with the Liouvillian, and (c) existence of a nonvanishing “entropy production” with respect to the intrinsic time.
Abstract Some basic concepts of the Prigogine quantum statistical theory of irreversible processes are discussed in connection with a previous paper by Chatzidimitriou‐Dreismann and Lippert. The removal of the “dual role” that the Hamiltonian plays within conventional quantum theory corresponds to the introduction of a star‐unitary operator 𝒯 called “intrinsic time.” The formalism of the previous paper is extended to systems that interact with external fields. A formal specification of 𝒯, furthermore, follows from the Lt invariance of the basic equations of motion and its connection with the intrinsic time. As an application to fluorescence spectroscopy, a photon counting experiment is presented. It allows us to detect certain predicted “dynamical” fluctuations in emission spectra. Some current experimental results are reported concerning the resolution of the broad fluorescence band of fluoranthene (dissolved in ethanol) through detection of the underlying finer vibrational structure.
Abstract Prigogine has emphasized that there is a “degeneracy” in quantum theory, where the Hamiltonian plays a dual role as (a) the generator of time evolution and (b) the energy observable. This prevents the exact treatment of irreversible phenomena on the microscopic quantum mechanical level. The removal of the aforementioned degeneracy can be achieved with the aid of a star‐unitary similarity transformation depending explicitly on the star‐Hermitian operator of the “intrinsic time”. Its physical significance is shown in the context of luminescence spectroscopy, where a novel kind of (dynamically induced or D‐) fluctuations around steady photoemission states is predicted. A photon counting/spectroscopic experiment has been performed which permits to detect D‐fluctuations and, at the same time, allows us to resolve broad spectroscopic bands by detection of the underlying finer “pure” vibronic transitions. The experimental findings clearly show the physical significance of Prigogine's Microscopic Theory.
We propose a novel approach to intrinsic decoherence without adding new assumptions to standard quantum mechanics. We generalize the Liouville equation just by requiring the dynamical semigroup property of time evolution and dropping the unitarity requirement. With no approximations and statistical assumptions we find a generalized Liouville equation which depends on two characteristic time t1 and t2 and reduces to the usual equation in the limit t1 = t2 -> 0. However, for t1 and t2 arbitrarily small but finite, our equation can be written as a finite difference equation which predicts state reduction to the diagonal form in the energy representation. The rate of decoherence becomes faster at the macroscopic limit as the energy scale of the system increases. In our approach the evolution time appears, a posteriori, as a statistical variable with a Poisson-gamma function probability distribution as if time evolution would take place randomly at average intervals t2 each evolution having a time width t1. This view point is supported by the derivation of a generalized Tam Mandelstam inequality. The relation with previous work by Milburn, with laser and micromaser theory and many experimental testable examples are described.
We propose a novel approach to intrinsic decoherence without adding new assumptions to standard Quantum Mechanics. We generalize the Liouville equation just by requiring the dynamical semigroup property of time evolution and dropping the unitarity requirement. With no approximations and specific statistical assumptions we find a generalized Liouville equation which depends on two characteristic time t1 and t2 and reduces to the usual equation in the limit t1 = t2 --> 0. However, for t1 and t2 arbitrarily small but finite, our equation can be written as a finite difference equation which predicts state reduction to the diagonal form in the energy representation. The rate of decoherence becomes faster at the macroscopic limit as the energy scale of the system increases. In our approach the evolution time appears, a posteriori, as a statistical variable as if time evolution would take place randomly at average intervals t2, each evolution having a time width t1. A generalized Tam Mandelstam inequality is derived. The relation with previous work by Milburn is discussed. The agreement with recent experiments on damped Rabi oscillations is described.
Every vertex of the magic cube of physics discussed so far introduced a major element in the development of physics and the whole of science. Mechanics introduced the basic concept of equations of motion, or time-evolution of a physical system given certain initial condition and the associated idea of determinism, even with its generalization in the form of chaos, the dependence of the outcomes of the temporal evolution on the initial conditions. These concepts taken for many particles, or elements of a physical system, led to the probabilistic description of many particle systems, in the realm of statistical physics and thermodynamics, whereby the averages over time of various observable quantities were related to the ensemble averages, that involve classical, so-called Kolmogorov, probability measures, at least for certain “equilibrium” situations. The second law of thermodynamics introduced the concept of entropy and its probabilistic increase, the first truly emergent, probabilistic law in physics (temperature and pressure, being other emergent quantities, apart from entropy, the measure of disorder of a statistical system). In the context of Newtonian gravity, one gets, at least in retrospect the geometric description of gravity in non-relativistic, Galilean spacetime. Galilean relativity, based on the asymmetric mixing of space and time, with the absolute, universal time, true for all observers, gets generalized in special theory of relativity to observer dependent space and observer dependent time, viewed as “slices” of a new concept of spacetime, with the new concept of causal structure, that distinguish timelike, null and spacelike events. This in turn is made into a full dynamical geometric structure of spacetime with a relativistic description of gravity stemming from curvature of spacetime, and a dynamical causal structure, tied to gravity. In our next story, the story of quantum theory, we will encounter fundamental, intrinsic probabilities, that can interfere, unlike the classical probabilities of statistical thermodynamics, and which are consistent with equations of motion of individual elements, and not, as in statistical thermodynamics, with temporal averages over the time evolution of many elements. This is the big new lesson of quantum theory, that transcends the lessons of classical spacetime physics.
We consider the temporal evolution of the entropy S of an isolated system from an initial non-equilibrium state 2· of entropy S = S i to the equilibrium state of maximum entropy S = S max > S i. The application of usual density matrix theory of quantum mechanics leads us to the surprising 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, because the entropy of the isolated system could not spontaneously increase from S i to S max by entropy production P > 0. This contradicts the second law of thermodynamics. We avoid this discrepancy by changing from quantum mechanics to quantum thermodynamics. Now the entropy production P = dS/dt and the time t are the relevant observables. In order to treat the temporal development of S and P, we at first prepare an initial state i of entropy production P i at time t < 0. Subsequently at ϑ t ≥ 0 the initial state i interacts with a continuum of final states f, which transfers the entropy production from i to f. The temporal evolution of the entropy production P(t) of our system is treated by means of the time-dependent perturbation theory. For that reason we split the operator of entropy production P = P 0 + W into two parts. The operator P 0 describes the unperturbed basic vectors of entropy production | i > and | f >, while the perturbation operator W determines the interaction between | i > and | f >. We expand the time-dependent state function < t | ψ > of the isolated system into the complete set of stationary eigenvectors | i > and | f > and calculate the time-dependent expansion coefficients c i(t) and c(f, t) by means of the dynamic equation of quantum thermodynamics − i k (d/dt) < t | ψ > = < t | P | ψ >. Here k means the Boltzmann constant, which is characteristic of quantum thermodynamics and represents the atomic entropy unit. The expansion coefficients are given by a system of coupled differential equations, from which we deduce an integral-differential equation for the coefficient c i(t) of the initial state i. Unfortunately it has been impossible to obtain an analytic solution of the complicated equation. However approximations are available. Accordingly, in the vicinity of the equilibrium state, the probability of still finding the initial state i of entropy production P i at time t is given by ω i(t) = | c i(t) |2 = exp (−Lt). The rate constant L ≥ σ depends on the density ρ(P f) of final states f of entropy production P f near by P i, and on the squared modulus of the perturbation matrix elements W if = < i | W | f >, which determines the interaction between | i > and | f >. The entropy production P(t) = P i ω i(t) = P i exp (−ϑLt) of the isolated system decreases exponentially and irreversibly with proceeding time t. Its rate-determining step is the transfer of entropy production from i to f. The integral over the entropy production of the final states f vanishes. Therefore the continuum states f do not contribute to the total entropy production P(t). The temporal evolution of the entropy S(t) can easily be calculated by integration of P(t). In conclusion, the entropy production of a non-equilibrium state is finite P(t) > 0 and only vanishes in the case of the equilibrium state P(∞) = 0. Thus, quantum thermodynamics agrees with the second law of thermodynamics.