@misc{oai:ir.soken.ac.jp:00004065,
 author = {岡澤, 晋 and オカザワ, ススム and OKAZAWA, Susumu},
 month = {2016-02-26, 2016-02-17},
 note = {We examine non-equilibrium aspects of the black hole thermodynamics by applying the non-equilibrium fluctuation theorems developed in the statistical physics. In particular, we consider a scalar field in a black hole background.
We derive the stochastic equations, i.e. the Langevin equation and the Fokker-Planck equations for a scalar field in a black hole background within the 0→ limit with the Hawking temperature πκ2/ fixed. By applying the fluctuation theorems to these effective equations of motion, we can derive the generalized second law of black hole thermodynamics, a linear response theorem of an energy flow and its non-linear generalizations as corollaries. We further investigate quantum corrections of the membrane paradigm.
We introduce infinitely many variables between the horizon and the stretched horizon and consider them as environmental variables. By integrating them, we can show that the variable at the stretched horizon behaves stochastically with a noise term. Though the environmental variables are living outside of the horizon, they can encode information in the black hole through choosing the Kruskal vacuum with the regularity condition at the horizon. In this sense, the integration of the environmental variables corresponds to integrating hidden variables in the horizon.
The system of the scalar field behaves stochastically due to the absorption of energy into the black hole and emission of the Hawking radiation from the black hole horizon. The dissipation comes from the classical causal property of the horizon; the black hole horizon absorbs matter and, once they fall in, they cannot come out. On the other hand, the noise term comes from the Hawking radiation, which is essentially quantum mechanical and, hence, we need to quantize the system in a black hole background in an appropriate way.
The thesis is organized as follows. In section 2, we briefly review the stochastic approach to thermodynamic systems, the Langevin equation and the Fokker-Planck equation. An important property of the stochastic equation is that it violates the time reversal symmetry which can be measured by an entropy increase in the path integral. In the next section 3, the fluctuation theorem for a stochastic system is reviewed. It relates the entropy increasing and decreasing probabilities. From the fluctuation theorem, the Jarzynski equality is derived. In addition, we explain the fluctuation theorem for a steady state and derivations of non-linear generalizations of the Green-Kubo formula. In section 4, we derive an effective stochastic equation of a scalar field in a black hole background. In
deriving the Langevin equation, the quantum property of the vacuum with the regularity condition at the horizon is very important, which is first explained. We then introduce a set of discretized equations of a scalar field near the black hole horizon, and integrate the variables between the horizon and the stretched horizon. The integration leads to an effective stochastic equation for a variable at the stretched horizon. This has the same spirit as deriving a Langevin equation of a system in contact with a thermal bath. In section 5, we apply the fluctuation theorem to a scalar field in a black hole background. We consider two different situations. In the first case, we put a scalar field and a black hole in a box with an insulating wall. By applying the fluctuation theorem, we can derive a relation connecting entropy decreasing probabilities with increasing ones. From this, the generalized second law of black hole thermodynamics can be derived. In the second case, the wall is assumed to be in contact with a thermal bath of a different temperature which is slightly lower than the Hawking temperature of the black hole. Then there is an energy flow from the black hole to the wall. By applying the steady state fluctuation theorem to it, a linear response theorem of an energy flow to the temperature difference and its non-linear generalizations can be obtained. In section 6, we extend the idea of the membrane paradigm. The equations of the classical membrane paradigm are essentially determined by the regularity condition. We further put the effect of the Hawking radiation to it.
In the appendix A, we review a derivation of the path integral form of the Fokker-Planck equation. In the appendix B, we review an example of the exact solution of the Fokker-Planck equation. In the appendix C, we will discuss the relation between the noise correlation and the flux of the Hawking radiation., 総研大甲第1588号},
 title = {Non-equilibrium Aspects of the Black Hole Thermodynamics},
 year = {}
}