Abstract
We consider a paradigmatic model of a quantum Brownian particle coupled to a thermostat consisting of harmonic oscillators. In the framework of a generalized Langevin equation, the memory (damping) kernel is assumed to be in the form of exponentially-decaying oscillations. We discuss a quantum counterpart of the equipartition energy theorem for a free Brownian particle in a thermal equilibrium state. We conclude that the average kinetic energy of the Brownian particle is equal to thermally-averaged kinetic energy per one degree of freedom of oscillators of the environment, additionally averaged over all possible oscillators’ frequencies distributed according to some probability density in which details of the particle-environment interaction are present via the parameters of the damping kernel.
Highlights
One of the enduring milestones of classical statistical physics is the theorem of the equipartition of energy [1], which states that energy is shared amongst all energetically-accessible degrees of freedom of a system and relates average energy to the temperature T of the system
Is described in Section 4; in Section 5, we convert the integro-differential Langevin equation into a set of differential equations; the equipartition theorem is discussed in Section 6; some selected physical regimes are analyzed in Section 7; we conclude the work with a brief résumé in Section 8; in Appendices A–C, we present some auxiliary calculations
It allows presenting an interesting interpretation of the quantum equipartition theorem: the average kinetic energy Ek of the Brownian particle is strongly related to the thermally-averaged kinetic energy Ek (ω ) per one degree of freedom of oscillators of the environment
Summary
One of the enduring milestones of classical statistical physics is the theorem of the equipartition of energy [1], which states that energy is shared amongst all energetically-accessible degrees of freedom of a system and relates average energy to the temperature T of the system. For average kinetic energy Ek (ω0 ) of a quantum harmonic oscillator of the eigenfrequency ω0 Equation (2) assumes the form: Ek = k B T It is the same expression as for the classical free particle. This means that for quantum systems, in contrast to a classical case, the average kinetic energy depends on the potential U ( x ), even in the weak coupling limit. We consider a free quantum particle coupled to its environment, which is modeled as a collection of harmonic oscillators of temperature T. This old clichéd system-environment model [3].
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
