TY - CHAP

T1 - Generalized Thermodynamics

AU - Matsoukas, Themis

N1 - Publisher Copyright:
© 2018, Springer Nature Switzerland AG.

PY - 2018

Y1 - 2018

N2 - The basis of all of our development up to this point has been the cluster ensemble, a discrete ensemble that generates every possible distribution of integers i with fixed zeroth and first order moments. Thermodynamics arises naturally in this ensemble when M and N become very large. In this chapter we will reformulate the theory on a mathematical basis that is more abstract and also more general. The key idea is as follows. If we obtain a sample from a given distribution h 0 , the distribution of the sample may be, in principle, any distribution h that is defined in the same domain. This sampling process defines a phase space of distributions h generated by sampling distribution h 0 . We will introduce a sampling bias via a selection functional W to define a probability measure on this space and obtain its most probable distribution. When the generating distribution h 0 is chosen to be exponential, the most probable distribution obeys thermodynamics. Along the way we will make contact with Information Theory, Bayesian Inference, and of course Statistical Mechanics.

AB - The basis of all of our development up to this point has been the cluster ensemble, a discrete ensemble that generates every possible distribution of integers i with fixed zeroth and first order moments. Thermodynamics arises naturally in this ensemble when M and N become very large. In this chapter we will reformulate the theory on a mathematical basis that is more abstract and also more general. The key idea is as follows. If we obtain a sample from a given distribution h 0 , the distribution of the sample may be, in principle, any distribution h that is defined in the same domain. This sampling process defines a phase space of distributions h generated by sampling distribution h 0 . We will introduce a sampling bias via a selection functional W to define a probability measure on this space and obtain its most probable distribution. When the generating distribution h 0 is chosen to be exponential, the most probable distribution obeys thermodynamics. Along the way we will make contact with Information Theory, Bayesian Inference, and of course Statistical Mechanics.

UR - http://www.scopus.com/inward/record.url?scp=85065825810&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85065825810&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-04149-6_7

DO - 10.1007/978-3-030-04149-6_7

M3 - Chapter

AN - SCOPUS:85065825810

T3 - Understanding Complex Systems

SP - 197

EP - 239

BT - Understanding Complex Systems

PB - Springer Verlag

ER -