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Fluctuations, Trajectory Entropy and Ziegler’s Maximum Entropy Production Principle

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Beyond the Second Law

Part of the book series: Understanding Complex Systems ((UCS))

Abstract

This chapter discusses two current interpretations of the maximum entropy production principle—as a physical principle and as an inference procedure. A simple model of relaxation of an isolated system towards equilibrium is considered for this purpose.

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Notes

  1. 1.

    According to their definitions [27–29], γ ii  > 0 and γ11γ22−γ 212  ≥ 0.

  2. 2.

    For small τ values: β ij Δα i Δα j  ≪ γ ij Δα i Δα * j (because γ ij  = γ 0 ij /τ).

  3. 3.

    For small τ values: dα * i /dt ≈ −Δα * i /τ. The minus sign arises from the fact that Δα * i is the difference between the initial and final value.

  4. 4.

    For one approximation, extremization is carried out for a random deviation from the equilibrium, whereas for the other approximation, it is carried out for a thermodynamic flux, i.e. the most probable deviation.

  5. 5.

    Mathematical models are absolutely unsuitable for falsification. So, a model is only some more or less crude and often one-sided reflection of some part of a phenomenon, whereas MEPP is the principle reflecting the dissipative properties that are observed in nature rather than in its model.

  6. 6.

    Indeed, the researcher’s intention to mathematically make the most unprejudiced prediction in the conditions of incomplete information about the system is the essence of this method. Therefore, if a phenomenon is very poorly experimentally studied (i.e. there are insufficient constraints), then anything can be predicted using MaxEnt (i.e. there are no truth criteria). In contrast, when MaxEP or MaxEnt are considered as physical principles, there are far fewer possibilities for drawing arbitrary conclusions. Methods that predict something specific for poorly studied phenomena (from which their falsifiability derives) are especially valuable.

  7. 7.

    If this cannot be achieved by selecting the constraints, then other kinds of informational entropy can always be used, for example, by Tsallis, Abe, Kullback, and many others [23, 30].

Abbreviations

t :

Time

A i (t):

Set of macroscopic state variables

A eq i :

Set of macroscopic state variables in equilibrium

W(α):

Probability of finding the equilibrium system in macrostate α

P(α 0 |α, τ):

Conditional probability that, given the system is in initial state α 0 , it will be in state α after time τ

S(α):

Entropy in state α

S tr(α 0 |α, τ):

Trajectory entropy that, given the system is in initial state α 0 , it will be in state α after time τ

ΔS(α):

S(α) − S(0)

X i :

Thermodynamic forces

J i :

Thermodynamic fluxes

L ij :

Kinetic coefficients

α i :

=A i (t) − A eq i

α :

Macroscopic (non-equilibrium) state vector with components α i

τ:

Observation time

Γ(α):

Number of microscopic states realizing macrostate α

Γ(α 0 |α, τ):

Number of microscopic trajectories realizing the transition from α 0 to α during time τ

Ω, Ξ:

Normalization constants

σ:

Entropy production (or entropy production density for the model under consideration)

β ij :

Coefficient that is inversely proportional to the variance of α relative to its equilibrium value (α = 0)

γ ij :

Coefficient that is inversely proportional to the variance of α relative to its average (most probable) value α*(α 0 , τ) during the transition from α 0 during time τ

γ 0 ij :

=γ ij τ

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Authors and Affiliations

  1. Ural Federal University, 19 Mira Street, Ekaterinburg, Russia, 620002

    Vladimir D. Seleznev & Leonid M. Martyushev

  2. Institute of Industrial Ecology, Russian Academy of Sciences, 20 S. Kovalevskaya Street, Ekaterinburg, Russia, 620219

    Leonid M. Martyushev

Authors
  1. Vladimir D. Seleznev
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  2. Leonid M. Martyushev
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Correspondence to Leonid M. Martyushev .

Editor information

Editors and Affiliations

  1. Australian National University, Research School of Biology, Canberra, Aust Capital Terr, Australia

    Roderick C. Dewar

  2. Australian National University, Research School of Astronomy and Astrophysics and the Research School of Earth Sciences, Canberra, Aust Capital Terr, Australia

    Charles H. Lineweaver

  3. School of Engineering and Information Technology, University of New South Wales at ADFA, Canberra, Aust Capital Terr, Australia

    Robert K. Niven

  4. School of Earth and Environment, University of Western Australia, CSIRO Earth Science and Resource Engineering, Crawley, West Australia, Australia

    Klaus Regenauer-Lieb

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Seleznev, V.D., Martyushev, L.M. (2014). Fluctuations, Trajectory Entropy and Ziegler’s Maximum Entropy Production Principle. In: Dewar, R., Lineweaver, C., Niven, R., Regenauer-Lieb, K. (eds) Beyond the Second Law. Understanding Complex Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40154-1_5

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