It is suggested that the nucleon mass arises largely as a self-energy of some primary fermion field through the same mechanism as the appearance of energy gap in the theory of superconductivity. The idea can be put into a mathematical formulation utilizing a generalized Hartree-Fock approximation which regards real nucleons as quasi-particle excitations. We consider a simplified model of nonlinear four-fermion interaction which allows a p5-gauge group. An interesting consequence of the symmetry is that there arise automatically pseudoscalar zero-mass bound states of nucleon-antinucleon pair which may be regarded as an idealized pion. In addition, massive bound states of nucleon number zero and two are predicted in a simple approximation.The theory contains two parameters which can be explicitly related to observed nucleon mass and the pion-nucleon coupling constant. Some paradoxical aspects of the theory in connection with the p5 transformation are discussed in detail.

Continuing the program developed in a previous paper, a "superconductive" solution describing the proton-neutron doublet is obtained from a nonlinear spinor field Lagrangian. We Gnd the pions of Gnite mass as nucleon-antinucleon bound states by introducing a small bare mass into the Lagrangian which otherwise possesses a certain type of the p5 invariance. In addition, heavier mesons and two-nucleon bound states are obtained in the same approximation. On the basis of numerical mass relations, it is suggested that the bare nucleon Geld is similar to the electron-neutrino Geld, and further speculations are made concerning the complete description of the baryons and leptons. ' Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345 (1961l; referred to hereafter as I. Y. Nambu, Proceedings of the 1960 A nnlal International Con ference on High-Energy Physics at

Stationary non-equilibrium states describe steady flows through macroscopic systems. Although they represent the simplest generalization of equilibrium states, they exhibit a variety of new phenomena. Within a statistical mechanics approach, these states have been the subject of several theoretical investigations, both analytic and numerical. The macroscopic fluctuation theory, based on a formula for the probability of joint space-time fluctuations of thermodynamic variables and currents, provides a unified macroscopic treatment of such states for driven diffusive systems. We give a detailed review of this theory including its main predictions and most relevant applications.

In this paper we formulate a dynamical fluctuation theory for stationary non equilibrium states (SNS) which covers situations in a nonlinear hydrodynamic regime and is verified explicitly in stochastic models of interacting particles. In our theory a crucial role is played by the time reversed dynamics. Our results include the modification of the Onsager-Machlup theory in the SNS, a general Hamilton-Jacobi equation for the macroscopic entropy and a non equilibrium, non linear fluctuation dissipation relation valid for a wide class of systems. The Boltzmann-Einstein theory of equilibrium thermodynamic fluctuations, as described for example in Landau-Lifshitz [1], states that the probability for a fluctuation from equilibrium in a macroscopic region of volume V is proportional to exp{V ∆S/k} where ∆S is the variation of entropy density calculated along a reversible transformation creating the fluctuation and k is the Boltzmann constant. This theory is well established and has received a rigorous mathematical formulation in classical equilibrium statistical mechanics via the so called large deviation theory [2]. The rigorous study of large deviations has been extended to hydrodynamic evolutions of stochastic interacting particle systems [3]. In a dynamical setting one may asks new questions, for example what is the most probable trajectory followed by the system in the spontaneous emergence of a fluctuation or in its relaxation to equilibrium. The Onsager-Machlup approach [4] answers precisely to this question: in the situation of a linear hydrodynamic equation, that is, close to equilibrium, the most probable emergence and relaxation trajectories are one the time reversal of the other. Developing the methods of [3], this theory has been extended to nonlinear regimes [5]. Onsager-Machlup assume the reversibility of the microscopic dynamics; however microscopically non reversible models were constructed where the above results still hold, [6,7].Emergence of large fluctuations, including OnsagerMachlup symmetry, has been observed in stochastically perturbed gradient type electronic devices [8]. In their work, these authors study also non gradient (i.e. non reversible) systems and observe violation of OnsagerMachlup symmetry.In the present paper we formulate a general theory of large deviations for irreversible processes, i.e. when detailed balance condition does not hold. This question was previously addressed in [10]. Natural examples are boundary driven stationary non equilibrium states (SNS), e.g. a thermodynamic system in contact with two reservoirs, but our theory covers, as a special case, also the model systems considered in [8]. In our approach a crucial role is played by the time reversed dynamics with respect to the stationary non equilibrium ensemble.Our results are: 1. The Onsager-Machlup relationship has to be modified: the emergence of a fluctuation takes place along a trajectory which is determined by the time reversed process.2. We show that the macroscopic entropy solves a Hamilton-Jacobi equation ge...