The Model of Quantum Thermodynamics From the First Principles: Quantum Halo or Small Environment

Abstract: The evolution of the joint system (JS) - ``quantum system (QS)+thermal bath (TB)" is considered in the framework of a complex probabilistic processes that satisfies the stochastic differential equation of the Langevin-Schr\"{o}dinger type. Two linearly coupled oscillators that randomly interact with the environment and with each other are selected as QS. In the case when the interactions obey the law of a white random process, all the construction of the statistical parameters of the QS and its environment are performed analytically in the form of double integrals and solutions of second-order partial differential equations. Expressions of time-dependent von Neumann entropy and its generalization are obtained, taking into account the self-organization and entanglement processes occurring in the JS. It is mathematically proved that as a result of the relaxation of JS in the TB, a small quantized environment is formed, which can be interpreted as a continuation of QS or its halo. Bell states formed as a result of the decay of coupled two linear oscillators are constructed taking into account the influence of the environment. The transitions between $(in)$ and $(out$) asymptotic states of QS are studied in detail taking into account the influence of TB. Within the framework of the model problem, the possibility of constructing quantum thermodynamics from the first principle is proved without using any additional conditions.