Abstract
We present a definition of equilibrium for Boltzmannian statistical mechanics based on the long-run fraction of time a system spends in a state. We then formulate and prove an existence theorem which provides general criteria for the existence of an equilibrium state. We illustrate how the theorem works with toy example. After a look at the ergodic programme, we discuss equilibria in a number of different gas systems: the ideal gas, the dilute gas, the Kac gas, the stadium gas, the mushroom gas and the multi-mushroom gas.
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Notes
- 1.
- 2.
At this point the measure need not be normalised.
- 3.
The dynamics is given by the evolution equations restricted to Z. We follow the literature by denoting it again by \(T_{t}\).
- 4.
We assume that \(\varepsilon \) is small enough so that \(\alpha (1-\varepsilon )> \frac {1}{2}\).
- 5.
We assume that \(\varepsilon <\gamma \).
- 6.
For further examples see Werndl and Frigg (2015a).
- 7.
The proof is given in Werndl and Frigg (2015a).
- 8.
It is allowed that the cells are of measure zero and that there are uncountably many of them.
- 9.
This example also shows that the largest macro-state need not be the equilibrium state: \(X^{\prime \prime }_{M_{w}}\) takes up \(1/2\) of \(X''\) and yet \(M_{w}\) is not the equilibrium state.
- 10.
In detail: \((Z,\Sigma _{Z},\mu _{Z},T_{t})\) is \(\varepsilon \)-ergodic, \(\varepsilon \in \mathbb {R},\,0\leq \varepsilon <1\), iff there is a set \(\hat {Z}\subset Z\), \(\mu _{Z}(\hat {Z})=1-\varepsilon \), with \(T_{t}(\hat {Z})\subseteq \hat {Z}\) for all t, such that the system \((\hat {Z},\Sigma _{\hat {Z}},\mu _{\hat {Z}},T_{t})\) is ergodic, where \(\Sigma _{\hat {Z}}\) and \(\mu _{\hat {Z}}\) is the \(\sigma \)-algebra \(\Sigma _{Z}\) and the measure \(\mu _{Z}\) restricted to \(\hat {Z}\). A system \((Z,\Sigma _{Z},\mu _{Z},T_{t})\) is epsilon-ergodic iff there exists a very small \(\varepsilon \) for which the system is \(\varepsilon \)-ergodic.
- 11.
Note that this is one of crucial differences between the dilute gas and the oscillator with a colour macro-variable of Sect. 10.4: the colour equilibrium does not depend on the system’s energy.
- 12.
The issue is the following Eq. (10.10) gives the distribution of largest size relative to the Lebesgue measure on the 6N-dimensional shell-like domain \(X_{ES}\) specified by the condition that \(E=\sum _{i=1}^{l}N_{i}e_{i}\). It does not give us the distribution with the largest measure \(\mu _{E}\) on the \(6N-1\) dimensional \(Z_{E}\). Strictly speaking nothing about the size of \(Z_{MB}\) (with respect to \(\mu _{E}\)) follows from the combinatorial considerations leading to Eq. (10.10). Yet it is generally assumed that the proportion of the areas corresponding to different distributions are the same on X and on \(X_{E}\) (or at least that the relative ordering is the same). Under that assumption \(Z_{E}\) is indeed the largest macro-region. We agree with Ehrenfest and Ehrenfest (1959, 30) that this assumption is in need of further justification, but grant it for the sake of the argument.
- 13.
One also think of the particles as bouncing back and forth in box. In this case the modulo of the momenta is preserved and a similar argument applies.
- 14.
In terms of the uniform measure on the momentum coordinates.
- 15.
Another possible treatment of the ideal gas is to consider the different macro-state structure given only by the coarse-grained position coordinates (i.e. the momentum coordinates are not considered). Then the effective dynamical system would coincide with the full dynamical system \((\varGamma ,\Sigma _{\varGamma },\mu _{\varGamma },T_{t})\). Relative to this dynamical system there would be an \(\gamma \)-0-equilibrium (namely the uniform distribution). That is, almost all initial conditions would spend most of the time in the macro-state that corresponds to the uniform distribution of the position coordinates.
- 16.
Speed, unlike velocity, is not directional and does not change when particle bounces off the wall.
- 17.
Bunimovich’s (1979) results are about one particle moving in a stadium-shaped box, but they immediately imply the results stated here about n non-interacting particles.
- 18.
It is assumed here that N is a multiple of l.
- 19.
The results in Bunimovich (2002) are all about one particle moving inside a mushroom-shaped box, but they immediately imply the results about a system of n non-interacting particles stated here.
- 20.
How small \(\varepsilon \) is depends on the exact shape of the box of the three elliptic mushrooms (it can be made arbitrarily small).
- 21.
Again, Bunimovich’s (2002) results are all about one particle moving inside a mushroom-shaped box, but they immediately imply the results about a system of n non-interacting particles stated here.
- 22.
This can always be arranged in this way—see Bunimovich (2002).
- 23.
By David Lavis and Reimer Kühn in private conversation.
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Werndl, C., Frigg, R. (2023). When Does a Boltzmannian Equilibrium Exist?. In: Soto, C. (eds) Current Debates in Philosophy of Science. Synthese Library, vol 477. Springer, Cham. https://doi.org/10.1007/978-3-031-32375-1_10
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