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Thermodynamics 2012 Part 7

Tham khảo tài liệu 'thermodynamics 2012 part 7', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Four Exactly Solvable Examples in Non-Equilibrium Thermodynamics of Small Systems 171 Heuristically the underlying time-inhomogeneous Markov process D t can be conceived as an ensemble of individual realizations sample paths . A realization is specified by a succession of transitions between the two states. If we know the number n of the transitions during a path and the times tkn 1 at which they occur we can calculate the probability that this specific path will be generated. A given paths yields a unique value of the microscopic work done on the system. For example if the system is known to remain during the time interval tk tk 1 in the ith state the work done on the system during this time interval is simply Ei tk 1 Ei tk . The probability of an arbitrary fixed path amounts at the same time the probability of that value of the work which is attributed to the path in question. Viewed in this way the work itself is a stochastic process and we denote it as W t . We are interested in its probability density p w t ỗ W t w where . denotes the average over all possible paths. We now introduce the augmented process W t D t which simultaneously reflects both the work variable and the state variable. The augmented process is again a time non-homogeneous Markov process. Actually if we know at a fixed time t both the present state variable j and the work variable w then the subsequent probabilistic evolution of the state and the work is completely determined. The work done during the time period t t where t t simply adds to the present work w and it only depends on the succession of the states after the time t . And this succession by itself cannot depend on the dynamics before time t1. The one-time properties of the augmented process will be described by the functions G w t1 w t limProb W t G w w e and D t i W t w and D t j j 7 imi 50 where i j 1 2. We represent them as the matrix elements of a single two-by-two matrix G w t w t Gij w t w t i G w t w t j . 51 We need an .