Single Trajectory Entropy Based On Trajectory Ensembles

The second law of thermodynamics shows that the entropy of macroscopic isolated systems never decreases and shows a obvious arrow of time.However,the microscopic particles that make up the macroscopic system follow a dynamic equation that is time reversible.So how realize the transition from microscopic reversibility to macroscopic irreversibility is a difficult task.This problem has been puzzling physicists for long time.To solve this problem,scientists has made lots of attempts.Scientists found that for the "small system",the evolution of some particles follows the second law of thermodynamics,while some particles violate the second law of thermodynamics.The ratio of the numbers of the latter to the numbers of the former is e-|?S|.This is known as the fluctuation theorem.To explore the evolution of particles that violate the second law of thermodynamics,it is necessary to define entropy at the level of the single trajectory produced by the evolution of particles.Therefore,the single trajectory entropy of particles is of great significance to the study of the reversible behavior of "small systems".In this thesis we study the single trajectory entropy of particles moving in bath 2001 Martens et al.put forward entangled trajectory method.In the entangled trajectory ensemble,trajectories are not independent but are "entangled" to each other.In this way,we can "track" a single trajectory and explore the physical mechanism shown by entropy evolution and fluctuation theoremThis thesis consists of four chapters,and it is organized as follows:In chapter one,we review the entropy and its development,single trajectory entropy and fluctuation theorem.In chapter two,we derive the entangled trajec-tory equations in the cases of one dimension and phase space.We also introduce the kernel density estimation theory and found single trajectory entropy theory of trajectory ensemble.In chapter three,in the case of one dimension we calculate the particle trajectories,single trajectory entropy,entropy production of the sin-gle trajectory and and entropy production of the medium in harmonic oscillator,stochastic resonance model and the triple-well potential model.For the one dimen-sion harmonic oscillator model,the particles localized symmetrically at two sides of the potential well obey the same law of motion.The system entropy calculated by entangled trajectory method agrees well with analytical result.For the stochastic resonance model,the diffusion constant will affect the mean square of displacement,which reflects the discretization degree is related to diffusion constant,then affect the single trajectory entropy.For the triple-well potential model,the entropy pro-duction of the medium is related to the trajectories of particles.In phase space,we calculate and analyze the trajectories of particles and single trajectory entropy in harmonic oscillator.We can find that the particles located at two sides of potential well symmetrically have the contrast law of motion,but they have the same change of single trajectory entropy.Regarded x?0 as a center,the particles located at the same side of the potential well have the same law of motion,but have the different change of single trajectory entropy.The chapter four is the conclusion of the paper and the outlook.