THERMODYNAMICS AND STATISTICAL MECHANICS
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1 Some Misconceptions about Entropy P 1 Some Misconceptions about Entropy P 2 THERMODYNAMICS AND STATISTICAL MECHANICS SOME MISCONCEPTIONS ABOUT ENTROPY Introduction the ground rules. Gibbs versus Boltmann entropies. That awful H-theorem. The Second Law of Thermodynamics. Tsallis and other heresies. Open the quantum box. Time asymmetry and the approach to equilibrium. Conclusions.
2 Some Misconceptions about Entropy P 3 Some Misconceptions about Entropy P 4 STATISTICAL MECHANICS GIBBS VS. BOLTZMANN INFERENCE A BAYESIAN PERSPECTIVE
3 Some Misconceptions about Entropy P 5 Some Misconceptions about Entropy P 6 INFERENCE AND STATISTICAL MECHANICS GIBBS VS. BOLTZMANN ENTROPIES
4 Some Misconceptions about Entropy P 7 Some Misconceptions about Entropy P 8 GIBBS VS. BOLTZMANN ENTROPIES II THAT AWFUL H-THEOREM IN ALL THE BOOKS
5 Some Misconceptions about Entropy P 9 Some Misconceptions about Entropy P 10 PROOF OF THE SECOND LAW OF THERMODYNAMICS THE SECOND LAW OF THERMODYNAMICS
6 Some Misconceptions about Entropy P 11 Some Misconceptions about Entropy P 12 THE THEORETICAL SECOND LAW ENTROPY INCREASE AND THE ARROW OF TIME Boltzmann s constant (k B = J K 1 ) is rather small, and the increase of phase-space volume is usually very large. (Taken from a Cambridge Part IB Physics Example Sheet.) Suppose that an infrared photon of energy 1 ev is absorbed by a dust grain at 300 K. The entropy increases by J K 1. This increases the available phase-space volume of the Universe by a factor of exp(38.7) = Scaled up to the size of this lecture theatre, we get a staggering exp(10 21 ) increase in volume PER SECOND. Macroscopic irreversibility is therefore not surprising in the least...
7 Some Misconceptions about Entropy P 13 Some Misconceptions about Entropy P 14 TSALLIS ENTROPIES AND OTHER HERESIES OPEN THE QUANTUM BOX There has been a lot of papers published concerning the Tsallis generalised entropy. These are one-parameter family of functions based on the q-derivative, and include the correct entropy as a special case q = 1. The entropic functions for other values of the parameter q are non-extensive (i.e. do not satisfy the Kangaroo axiom). Quantum particle in ground state of 1-dimensional box 1 2 a < x < 1 2a. At t = 0 the box is opened to a < x < a. We can expand the old ground state in terms of the new states. It is still in a pure state, though not an energy eigenstate, and the entropy is still zero. The maximised entropies for q 1 are not equal to the experimental entropy defined by Clausius. The industry of q-generalisations has taken hold in many different fields. It is mathematically self-consistent, and may be great fun, but its track record for concrete achievements is (in my opinion) still zero. That said, Constantino Tsallis has published a very impressive list of applications throughout physics and astrophysics. But I don t believe him... Evolution of box wavefunction after a long time. The probability either has a central hump, or two symmetrical ones, as the wavefunction displays the interference of the 2 lowest frequency modes.
8 Some Misconceptions about Entropy P 15 Some Misconceptions about Entropy P 16 OPEN THE QUANTUM BOX III OPEN THE QUANTUM BOX II If the box is opened one-sidedly the oscillations are more violent. The averaged distribution has two humps and the phase-averaged entropy increase is A movie shows that the particle oscillates around enjoying its new-found freedom. It doesn t remotely settle down. Some authors average the phases and thereby get an entropy increase of The time average (500 samples) of the probability (red line) is almost indistinguishable from the phase average. But this is NOT thermodynamics.
9 Some Misconceptions about Entropy P 17 Some Misconceptions about Entropy P 18 OPEN THE QUANTUM BOX IV TIME ASYMMETRY IN PHYSICS The canonical distribution with the same average energy has an entropy of If we impose symmetry on the wavefunction the entropy increase is and there is a hump in the middle.
10 Some Misconceptions about Entropy P 19 Some Misconceptions about Entropy P 20 TIME ASYMMETRY AND NON-EQUILIBRIA BROWNIAN MOTION
11 Some Misconceptions about Entropy P 21 Some Misconceptions about Entropy P 22 BROWNIAN MOTION AND TIME ASYMMETRY EQUILIBRIUM ENSEMBLE FOR BROWNIAN MOTION
12 Some Misconceptions about Entropy P 23 Some Misconceptions about Entropy P 24 UNCERTAINTY VERSUS FLUCTUATIONS CONCLUSIONS
13 Some Misconceptions about Entropy P 25 Some Misconceptions about Entropy P 26 APPENDIX PROOF OF S G = S E APPENDIX PROOF OF S G = S E II
14 Some Misconceptions about Entropy P 27 BROWNIAN MOTION
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