Abstract
It is generally observed that if a dynamical system is sufficiently complex, then as time progresses it will share out energy and other properties amongst its component parts to eliminate any initial imbalances, retaining only fluctuations. This is known as energy dissipation and it is closely associated with the concept of thermodynamic irreversibility, measured by the increase in entropy according to the second law. It is of interest to quantify such behaviour from a dynamical rather than a thermodynamic perspective and to this end stochastic entropy production and the time-integrated dissipation function have been introduced as analogous measures of irreversibility, principally for stochastic and deterministic dynamics, respectively. We seek to compare these measures. First we modify the dissipation function to allow it to measure irreversibility in situations where the initial probability density function (pdf) of the system is asymmetric as well as symmetric in velocity. We propose that it tests for failure of what we call the obversibility of the system, to be contrasted with reversibility, the failure of which is assessed by stochastic entropy production. We note that the essential difference between stochastic entropy production and the time-integrated modified dissipation function lies in the sequence of procedures undertaken in the associated tests of irreversibility. We argue that an assumed symmetry of the initial pdf with respect to velocity inversion (within a framework of deterministic dynamics) can be incompatible with the Past Hypothesis, according to which there should be a statistical distinction between the behaviour of certain properties of an isolated system as it evolves into the far future and the remote past. Imposing symmetry on a velocity distribution is acceptable for many applications of statistical physics, but can introduce difficulties when discussing irreversible behaviour.
Highlights
Thermodynamic irreversibility refers to the fact that most physical processes that take place in the world tend to evolve in a preferred direction as time goes by
Irreversible behaviour remains a puzzle today, over 150 years after it was first quantified using the concept of thermodynamic entropy production
If irreversibility is a consequence of complex but time-reversible dynamics why does past behaviour not resemble the predictions of development in the future? How is irreversibility to be made distinct from the formation and dissolution of patterns of the kind shown in figure 1? In the simplest presentation of Boltzmann’s ideas, it is difficult to avoid the conclusion that entropy is currently at a minimum, quite at odds with reality
Summary
Measures of thermodynamic irreversibility in deterministic and stochastic dynamics Ian J Ford. Share out energy and other properties amongst its component parts to eliminate any initial imbalances, Any further distribution of this work must maintain retaining only fluctuations This is known as energy dissipation and it is closely associated with the attribution to the concept of thermodynamic irreversibility, measured by the increase in entropy according to the author(s) and the title of the work, journal citation second law. First we modify the dissipation function to allow it to measure irreversibility in situations where the initial probability density function (pdf) of the system is asymmetric as well as symmetric in velocity We propose that it tests for failure of what we call the obversibility of the system, to be contrasted with reversibility, the failure of which is assessed by stochastic entropy production. Imposing symmetry on a velocity distribution is acceptable for many applications of statistical physics, but can introduce difficulties when discussing irreversible behaviour
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