Two-Way Thermodynamics
A model in which two weakly coupled systems maintain opposite running thermodynamic arrows of time is exhibited. Each experiences its own retarded electromagnetic interaction and can be seen by the other. “Time” is thus a statistical concept. Paradoxes ar
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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Some Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Opposite Arrows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix: Precise Definition of the Modified “Cat” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Abstract A model in which two weakly coupled systems maintain opposite running thermodynamic arrows of time is exhibited. Each experiences its own retarded electromagnetic interaction and can be seen by the other. “Time” is thus a statistical concept. Paradoxes are also explored from the standpoint of initial and final value problems.
Keywords Time · Statistical mechanics · Model systems · Paradoxes
L. S. Schulman () Physics Department, Clarkson University, Potsdam, NY, USA e-mail: [email protected] © Springer Nature Switzerland AG 2020 B. Sriraman (ed.), Handbook of the Mathematics of the Arts and Sciences, https://doi.org/10.1007/978-3-319-70658-0_108-1
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L. S. Schulman
Introduction The arrow of time is perceived by all of us and is made precise by the second law of thermodynamics. Let me explain both assertions of the previous sentence: The perception is simple. The past is fixed and often known; the future is unpredictable and generally unknown. This is an arrow appreciated by all. Now a precise version. The “second law of thermodynamics,” is the statement that entropy increases or stays constant as time moves forward. Entropy has a practical and a theoretical meaning. Its practical meaning is the amount of heat transferred divided by the temperature at which the transfer took place. Thus the entropy increase of a gram of ice that melts is 334 joules (about 80 calories, what it takes to melt the ice), divided by T (about 273 ◦ K), where entropy has units of joules per Kelvin (and Kelvin is a measure of temperature above absolute zero). It is thus about 1.22 joules per Kelvin for 1 gram of H2 O at standard pressure. The theoretical meaning of entropy is both as the logarithm of multiplicity (divided by kBoltzmann 1 ) and as missing information. It can be shown that all definitions of entropy agree (up to a multiplicative constant), and in the following, I’l
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