### Abstract

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.

Original language | English (US) |
---|---|

Title of host publication | Understanding Complex Systems |

Publisher | Springer Verlag |

Pages | 197-239 |

Number of pages | 43 |

DOIs | |

State | Published - Jan 1 2018 |

### Publication series

Name | Understanding Complex Systems |
---|---|

ISSN (Print) | 1860-0832 |

ISSN (Electronic) | 1860-0840 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Software
- Computational Mechanics
- Artificial Intelligence

### Cite this

*Understanding Complex Systems*(pp. 197-239). (Understanding Complex Systems). Springer Verlag. https://doi.org/10.1007/978-3-030-04149-6_7

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*Understanding Complex Systems.*Understanding Complex Systems, Springer Verlag, pp. 197-239. https://doi.org/10.1007/978-3-030-04149-6_7

**Generalized Thermodynamics.** / Matsoukas, Themis.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

TY - CHAP

T1 - Generalized Thermodynamics

AU - Matsoukas, Themis

PY - 2018/1/1

Y1 - 2018/1/1

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 -