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The delay Lotka-Volterra and delay Toda lattice equations are delay-differential extensions of the well-known soliton equations, the Lotka-Volterra and Toda lattice equations, respectively. This paper investigates integrable properties of the delay Lotka-Volterra and delay Toda lattice equations, and study the relationships to the already known delay Painleve equations. First, Backlund transformations, Lax pairs and an infinite number of conserved quantities of these delay soliton equations are constructed. Then, applying spatial 2-periodic reductions to them, we show the known delay Painleve equations are derived. Using these reductions, we construct the N-soliton-type determinant solutions of the autonomous versions of delay Painleve equations, and the Casorati determinant solution of a higher order analogue of the discrete Painleve II equation., Comment: 32 pages

A delay analogue of the box and ball system (BBS) is presented. This new soliton cellular automaton is constructed by the ultra-discretization of the delay discrete Lotka-Volterra equation, which is an integrable delay analogue of the discrete Lotka-Volterra equation. Soliton patterns generated by this delay BBS are classified into normal solitons and abnormal solitons. Normal solitons have a clear relationship to the solitons of the BBS with K kinds of balls. On the other hand, abnormal solitons show various types of novel soliton patterns, which have not been observed in almost all known BBSs. We obtain them by numerical experiments, and then construct {\tau}-functions of them analytically in 1-soliton cases.

For classical systems driven out of equilibrium, Crooks derived a relation (the Crooks-Jarzynski relation), whose special cases include a relation (the Crooks relation) equivalent to the Kawasaki non-linear response relation. We derive a quantum extension of the Crooks-Jarzynski relation without explicitly using the principle of microreversibility. Its special cases lead to the Jarzynski equality and the standard linear response theory with a Green-Kubo formula with a canonical correlation function. We also derive a quantum extension of the Crooks relation using the principle of microreversibility. Its special cases lead to the Jarzynski equality, the Crooks transient fluctuation theorem, and the fluctuation theorem for current or shear stress, which leads to a Green-Kubo formula with a symmetrized correlation function. For each quantum Crooks relation, there exists a corresponding quantum Crooks-Jarzynski relation. Using either relation, we can derive the Jarzynski equality, the fluctuation theorems mentioned above, and the standard linear response theory., Comment: 39 pages. Revised to provide the correct relationship between the quantum Crooks relation and its corresponding quantum Crooks-Jarzynski relation. Their relationship described in the previous versions of this manuscript was incorrect

This paper has been withdrawn by the author. For the reason, see the bottom paragraph of this abstract. By generalizing Tasaki's work on the second law of thermodynamics for an adiabatic process between two equilibrium states of a macroscopic quantum compound system, we obtain an extension of the second law to a transient adiabatic process that takes a macroscopic quantum compound system consisting of a system of interest and two heat reservoirs from an initial equilibrium state to a final non-equilibrium state, where the system of interest is in a steady heat conduction state. For the system of interest, we define its generalized entropy so that it becomes an experimentally measurable quantity and exactly satisfies an extension of the Clausius relation. This paper has been withdrawn by the author because the generalized entropy presented in this article violates the second law when the system in a steady heat conduction state is detached from the two reservoirs and then attached to a reservoir at the temperature T^bar defined in Sec.1.2 so that it will reach an equilibrium state at T^bar. In this irreversible process, the total entropy for the system and the reservoir remains constant and thus violates the second law., Comment: This paper has been withdrawn by the author because the generalized entropy presented in this article violates the second law. For more detail, see the bottom paragraph in the abstract

For a quantum system in a steady state with a constant current of heat or particles driven by a temperature or chemical potential difference between two reservoirs attached to the system, the fluctuation theorem for the current was previously shown to lead to the Green-Kubo formula for the linear response coefficient for the current expressed in terms of the symmetrized correlation function of the current density operator. In this article, we show that for a quantum system in a steady state with a constant rate of work done on the system, the fluctuation theorem for a quantity induced in the system also leads to the Green-Kubo formula expressed in terms of the symmetrized correlation function of the induced quantity. As an example, we consider a fluid in a steady shear flow driven by a constant velocity of a solid plate moving above the fluid., Comment: 34 pages. Minor corrections and changes. An expanded version of Sec.XIV as well as Sec.IX through Sec.XII of arXiv:1104.5190

For a deeply supercooled liquid near its glass transition temperature, we suggest a possible way to connect the temperature dependence of its molar excess entropy to that of its viscosity by constructing a macroscopic model, where the deeply supercooled liquid is assumed to be a mixture of solid-like and liquid-like micro regions. In this model, we assume that the mole fraction x of the liquid-like micro regions tends to zero as the temperature T of the liquid is decreased and extrapolated to a temperature Tg*, which we assume to be below but close to the lowest glass transition temperature Tg attainable with the slowest possible cooling rate for the liquid. Without referring to any specific microscopic nature of the solid-like and liquid-like micro regions, we also assume that near Tg, the molar enthalpy of the solid-like micro regions is lower than that of the liquid-like micro regions. We then show that the temperature dependence of x is directly related to that of the molar excess entropy. Close to Tg, we assume that an activated motion of the solid-like micro regions controls the viscosity and that this activated motion is a collective motion involving practically all of the solid-like micro-regions so that the molar activation free energy for the activated motion is proportional to the mole fraction, 1-x, of the solid-like micro regions. The temperature dependence of the viscosity is thus connected to that of the molar excess entropy through the temperature dependence of the mole fraction x. As an example, we apply our model to a class of glass formers for which the molar excess entropy at temperatures near Tg is proportional to 1-T/TK with TK < Tg \sim Tg* and find their viscosities to be well approximated by the Vogel-Fulcher-Tamman equation for temperatures very close to Tg. We estimate the values of three parameters in our model for three glass formers in this class., Comment: 29 pages. Extensively revised in sections I, II, III.G, III.H, IV, VI, and VII

For the four-dimensional SU(N) Euclidean Yang-Mills theory in the Landau gauge, we present two sets of gauge field configurations that satisfy the Euclidean equations of motion. These configurations generate four-dimensional chromo-field flux sheets whose spatial cross sections are three-dimensional chromo-field flux tubes. In lattice simulations, they may be detected as center vortices. The first set of gauge field configurations generates chromo-electric flux tubes that should contribute to a chromo-electric flux tube between two static color charges. The string tension for two static color charges in representation r then naturally satisfies the Casimir scaling. Applying a gauge transformation to this set of gauge field configurations, we can transform them into those in the maximal Abelian gauge. These transformed configurations generate chromo-electric flux tubes that should contribute to those observed between two static quarks in lattice simulations performed in the maximal Abelian gauge. The second set of gauge field configurations generates chromo-magnetic flux tubes. When rotated in a plane that includes the temporal-axis and is perpendicular to the flux tube axis, the rotated gauge field configuration generates a chromo-electric flux tube and should contribute to the chromo-electric flux tubes observed in lattice simulations in the Landau gauge. We also argue that when regulated on a lattice, any of the flux sheet gauge field configuration with a finite flux sheet thickness is located on the Gribov horizon in the infinite lattice volume limit. We thus suggest that these sets of gauge field configurations contribute significantly to the low energy properties of QCD, particularly the quark confinement., Comment: 31 pages. The version to appear in Physical Review D

For a deeply supercooled liquid just above its glass transition temperature, we present a simple thermodynamic model, where the deeply supercooled liquid is assumed to be a mixture of solid-like and liquid-like micro regions. The mole fraction of the liquid-like regions controls the thermodynamic properties of the supercooled liquid while that of the solid-like regions controls its relaxation time or viscosity. From the universal temperature dependence of the molar excess entropy for the deeply supercooled liquids, we derive the temperature dependence of the mole fraction of the liquid-like regions to obtain the universal temperature dependence of the relaxation time or the viscosity for the deeply supercooled liquids. A central parameter of our model is then shown to be a measure for the fragility of a supercooled liquid. We also suggest a way to test our physical picture of deeply supercooled liquids by means of molecular dynamics simulations of model liquids., Comment: This paper has been withdrawn by the author as it is superseded by arXiv:1202.0918

We extend a classical relation by Crooks to quantum systems and show that it unifies the Crooks transient fluctuation theorem and the Kawasaki non-linear response relation, which leads to the standard linear response theory. We also show that the Green-Kubo formula for a system in a steady state driven by a time-independent driving force is always expressed in terms of the symmetrized correlation function for a quantity induced by the driving force and that a generalized entropy for systems in steady heat conduction states satisfies extensions of the Clausius and the Gibbs relations exactly., Comment: 31 pages. Sec.III, Sec.IV, and Sec.IX have been substantially revised. The results remain mainly intact