The kinetic uncertainty relation (KUR) is a trade-off relation between the precision of an observable and the mean dynamical activity in a fixed time interval for a time-homogeneous and continuous-time Markov chain. In this letter, we derive the KUR on the first passage time for the time-integrated current from the information inequality at stopping times. The relation shows that the precision of the first passage time is bounded from above by the mean number of jumps up to that time. We apply our result to simple systems and demonstrate that the activity constraint gives a tighter bound than the thermodynamic uncertainty relation in the regime far from equilibrium.

We study fluctuations of pressure in equilibrium for classical particle systems. In equilibrium statistical mechanics, pressure for a microscopic state is defined by the derivative of a thermodynamic function or, more mechanically, through the momentum current. We show that although the two expectation values converge to the same equilibrium value in the thermodynamic limit, the variance of the mechanical pressure is in general greater than that of the pressure defined through the thermodynamic relation. We also present a condition for experimentally detecting the difference between them in an idealized measurement of momentum transfer.

Emergence of deterministic and irreversible macroscopic behavior from deterministic and reversible microscopic dynamics is understood as a result of the law of large numbers. In this paper, we prove on the basis of the theory of algorithmic randomness that Martin-Löf random initial microstates satisfy an irreversible macroscopic law in the Kac infinite chain model. We find that the time-reversed state of a random state is not random as well as violates the macroscopic law.A Turing machine is a special-purpose machine in the sense that the machine computes one computable function. Since we can code a program, which is just a string, by a natural number in a computable manner, we can construct a universal Turing machine, which is a model of the present-day computers. This is why today we can implement any program by using only one computer.Theorem 2.1.4 (universal Turing machine) There is a partial computable function of two variables g such that g(e, x) = f e (x) for any input x and any partial computable function f e indexed by a natural number e.

For a classical system consisting of N-interacting identical particles in contact with a heat bath, we define the free energy from thermodynamic relations in equilibrium statistical mechanics. Concretely, the temperature dependence of the free energy is determined from the Gibbs-Helmholtz relation, and its volume dependence is determined from the condition that the quasi-static work in a volume change is equal to the free energy change. Now, we argue the free energy difference in a quasi-static decomposition of small thermodynamic systems. We can then determine the N dependence of the free energy, which includes the Gibbs factorial N! in addition to the phase space integration of the Gibbs–Boltzmann factor.

We propose a prequential or sequentially predictive formulation of the work extraction where an external agent repeats the extraction of work from a heat engine by cyclic operations based on his predictive strategy. We show that if we impose the second law of thermodynamics in this situation, the empirical distribution of the initial microscopic states of the engine must converge to the Gibbs distribution of the initial Hamiltonian under some strategy, even though no probability distribution are assumed. We also propose a protocol where the agent can change only a small number of control parameters linearly coupled to the conjugate variables. We find that in the restricted situation the prequential form of the second law of thermodynamics implies the strong law of large numbers of the conjugate variables with respect to the control parameters. Finally, we provide a game-theoretic interpretation of our formulation and find that the prequential work extraction can be interpreted as a testing procedure for random number generator of the Gibbs distribution.

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