Quantum Statistical Ensemble Resilient to Thermalization

Abstract

The sampling of the wave function within a suitable ensemble is an important tool in the statistical analysis of a molecule interacting with its environment. The uniform statistical distribution of quantum pure states in an active space is often the privileged choice. However, such a distribution with constant average populations of eigenstates is not preserved upon the interaction between quantum systems. This appears as a severe methodological shortcoming, as long as a quantum system can be always considered as the result of interactions among previously isolated subsystems. In the present work we formulate an alternative statistical ensemble of pure states that is robust with respect to interaction, and it is thus preserved when subsystems are merged. It is derived from the condition of invariance of the average populations upon interaction between quantum systems in the same thermal state. These average populations allow a simple identification of the thermodynamic properties of the system. We find that such a statistical distribution is robust with respect to interaction of systems at different temperatures reproducing the thermalization of macroscopic bodies, and for this reason we identify it as the Thermalization Resilient Ensemble.

Thermal Pure States for Finite and Isolated Quantum Systems

“…We emphasize that the TRE ensemble is specifically designed to study the relaxation dynamics and the energy flow in quantum systems that are brought in thermal contact. , Indeed, the TRE allows the preparation of two initially isolated systems, each characterized by a well-defined temperature, and predicts the thermal state of the joined system based on the invariance of the TRE average populations upon thermal contact . This a priori knowledge of the thermal state of the final system gives a well-defined reference to evaluate whether the system thermalizes under its own unitary dynamics or localization phenomena prevent reaching the thermal state. , …”

“…To describe an ensemble of pure states we introduce the probability density p ( P ) on the set of populations with normalization where d P = d P 1 , d P 2 , ..., d P N and P k ≥ 0 as the integration domain of each population under the normalization condition. The ensemble average of a generic function f ( P ) of the population set P is then calculated as Our objective is formulating a probability density p ( P ) under the constraint that the corresponding average populations are exactly the thermalization resilient ensemble populations derived in ref , namely The parameter ζ ≥ 0 is used to classify the different thermal states of the system with the two limiting cases of ζ = 0 for infinite temperature when all the quantum states have the same average population and of ζ → ∞ for the zero-temperature limit when only the ground state is populated. Moreover, ζ is the control parameter of the statistic, meaning that different population distributions p ζ ( P ) are assigned to the isolated system according to the values of the thermal parameter ζ.…”

“…If we assume that the state of each system prior to interaction is a random pure state of a given subspace of the Hilbert space, the state of the composite system will not be any more a random pure state. By imposing the condition that the functional dependence of the average populations on the energy is invariant upon the interaction between two identical systems in the same thermal state, the average populations of TRE were derived . They are characterized by an exponential dependence on the energy in analogy with Boltzmann populations.…”

“…By imposing the condition that the functional dependence of the average populations on the energy is invariant upon the interaction between two identical systems in the same thermal state, the average populations of TRE were derived. 34 They are characterized by an exponential dependence on the energy in analogy with Boltzmann populations. Such a canonicallike statistic of the average populations results exclusively from the requirement of invariance of the average population upon contact of two systems, and it is therefore valid for the isolated system without invoking the interaction with a thermal bath.…”

“…Despite the simplicity of the statistic of random pure states, they are not suitable to define the thermal state of a finite isolated quantum system because of point (b) above. In particular, because of the discreteness of the energy spectrum, the entropy is not a continuous function of the energy, which prevents a consistent definition of temperature. − An alternative ensemble of pure states has been derived by us in ref under the name of thermalization resilient ensemble (TRE). A distinctive feature of the TRE is that it allows a full thermodynamic characterization of finite isolated quantum systems with discrete energy spectra avoiding the problem of the discontinuity of the entropy function. ,,, The average populations of the thermalization resilient ensemble have the property of being practically invariant upon thermal contact between two quantum systems that are initially independent.…”