The Interaction with the Environment

In statistical mechanics, there exists a commonly accepted postulate according to which a “small” system interacting with a “large” system (a heat bath), which stays in the state of statistical equilibrium, eventually approaches the state of statistical equilibrium itself. Mitropolskii and Petrina (Ukr Math J 45:171–214, 1993) and (Elementary example for establishing statistical equilibrium in a system coupled to a thermostat. On some statistical methods in mathematical physics (in Russian). Publ Acad Sci Ukr SSR Kiev, pp 115–137, 1945) rigorously proved the validity of this postulate for the first time. As the system and the heat bath, he considered an oscillator linearly coupled to a collection of harmonic oscillators, simulating the bath in which the single oscillator is immersed. However, without explicitly working out the statistical properties of the fluctuations. These properties, together with a quantum mechanical transcription of the model, has been accomplished by Magalinskiĭ (Sov Phys JETP 9:1381. JETP 36, 1942, 1959). Is this model we will unwrap in this chapter and we apply it to understand a quantum ratchet in the quantum range. For an excelent review of the evolution of the quantum Langevin equation, see P. Reimann (Phys Rep 361:57–265, 2002).