Evolution equations summary

The plan of this chapter is the following. Section II gives a summary of the phenomenology of irreversible processes and set up the stage for the results of nonequilibrium statistical mechanics to follow. In Section III, it is explained that time asymmetry is compatible with microreversibility. In Section IV, the concept of Pollicott-Ruelle resonance is presented and shown to break the time-reversal symmetry in the statistical description of the time evolution of nonequilibrium relaxation toward the state of thermodynamic equilibrium. This concept is applied in Section V to the construction of the hydrodynamic modes of diffusion at the microscopic level of description in the phase space of Newton s equations. This framework allows us to derive ab initio entropy production as shown in Section VI. In Section VII, the concept of Pollicott-Ruelle resonance is also used to obtain the different transport coefficients, as well as the rates of various kinetic processes in the framework of the escape-rate theory. The time asymmetry in the dynamical randomness of nonequilibrium systems and the fluctuation theorem for the currents are presented in Section VIII. Conclusions and perspectives in biology are discussed in Section IX. 

Figure 2.14. Evolution with temperature of the full width y0 at half maximum of the 0-0 absorption peak. Hollow circles represent our results from Kramers-Kronig analysis, (a) Evolution between 0 and 77 K. The solid line was drawn using equation (2.126) and adjusted parameters y, =72cm 1, hfi = 27cm" . The dashed line connects the results of our model (2.127)—(2.130) for six different temperatures, (b) Evolution between 0 and 300 K. The full circles are taken from ref. 62. This summary of the experimental results shows the linear behavior between 30 and 50 K., and the sublinear curvature at temperatures above 200 K. 

Summary. The synthesis of new elements takes place inside stars. How do stars evolve and distribute this creation to the universe at large This article starts with the observables that the theory of stellar evolution aims to reproduce, and gives a quick overview of what that theory predicts (Sects. 2-3). It presents the equations governing stellar structure and evolution (Sects. 4-6) and the physics of stellar interiors (Sects. 7-9). Approximate and numerical methods for their solution are outlined (Sects. 10-11) and the general results of stellar structure and evolution are discussed (Sects. 12-13). The structure and evolution of horizontal-branch stars, hydrogen-deficient stars and other stelfar remnants are also considered (Sects. 14-15). 

Summary. In this chapter, a simplified method for mapping objects based on the level set method is introduced. Level set and marching methods are used to map connected volumes within 3D seismic data. The simpler marching method solves the stationary problem stated by the level set formulation. The evolution of the object, from a seed point to the boundary, is described by a differential equation. 

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