Abstract
Simulations have played a key role in the recent studies on nonequilibrium. And simulations operate on computers to obtain solutions of equations in phase space: therefore phase space points are given a digital representation which might be very precise but rarely goes beyond 32 bits per coordinate.
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Notes
- 1.
Naive expectation would have been that if the manifold \(\varXi \) and the map \(S\) are smooth, say \(C^\infty \), also \(V^\alpha (x)\) and \(W^\alpha (x)\) depend as smoothly on \(x\).
- 2.
Notice that under the chaotic hypothesis motions on the attracting sets \(\mathcal {\mathcal{A}}\) are Anosov systems so that the stable and unstable manifolds of every point are dense: therefore \(W^s(x)\cap B_\gamma (x)\) is a dense family of layers in \(B_\gamma (x)\), but only one is connected and contains \(x\), if \(\gamma \) is small enough.
- 3.
The exception is associated with points \(x\) which are on the boundaries of the rectangles or on their iterates. In such cases it is possible to assign the symbol \(\xi _0\) arbitrarily among the labels of the rectangles to which \(x\) belongs: once made this choice a compatible history \({{\varvec{\xi }}}\) determining \(x\) exists and is unique.
- 4.
Formally let \(E_i\in \mathcal {\mathcal{P}}\), \(x\in E_i\) and \(\delta (x)=E_i\cap W_u(x)\): then if \(M_{i,j}=1\), i.e. if the interior of \(SE_i\) visits the interior of \(E_j\), it is \(\delta (Sx)\subset S\delta (x)\).
- 5.
The map is not obtainable as a Poincaré’s section of the orbits of a 3-dimensional manifold simply because its Jacobian determinant is not \(+1\).
- 6.
The name is chosen to mark the distinction with respect to the parallelepipeds of the coarse partition.
- 7.
Here it is essential that the chaotic hypothesis holds, i.e. that the system is hyperbolic, otherwise if the system has long time tails the analysis becomes much more involved and so far it can be dealt, even if only qualitatively, on a case by case basis.
- 8.
To get an idea of the orders of magnitude consider a rarefied gas of \(N\) mass \(m\) particles of density \(\varrho \) at temperature \(T\): the metric on phase space will be \(ds^2=\sum _i(\frac{d\mathbf{p}_i^2}{m k_B T}+\frac{d\mathbf{q}_i^2}{\varrho ^{-2/3}})\); each coarse cell will have size at least \(\sim \sqrt{mk_B T}\) in momentum and \(\sim \varrho ^{-\frac{1}{3}}\) in position; this is the minimum precision required to give a meaning to the particles as separate entities. Each microcell could have coordinates represented with \(32\) bits will have size of the order of \(\sqrt{mk_B T}2^{-32}\) in momentum and \(\varrho ^{-\frac{1}{3}}2^{-32}\) in position and the number of theoretically possible phase space points representable in the computer will be \(O((2^{32})^{6N})\) which is obviously far too large to allow anything being close to a recurrence in essentially any simulation of a chaotic system involving more than \(N=1\) particle.
- 9.
With extreme care it is sometimes, and in equilibrium, possible to represent a chaotic evolution \(S\) with a code \(\overline{S}\) which is a true permutation: the only example that I know, dealing with a physically relevant model, is in [14] .
- 10.
However in [1, #42] the \(w\)’s denote integers rather than phase space volumes.
- 11.
Simply because \(q_j\) have finitely many values, \(W\) is fixed and the angle \(\varphi =\varphi (x)\) between stable and unstable manifolds at \(x\) is bounded away fro \(0,\pi \) because of the transversality of the manifolds (in Anosov maps).
- 12.
Informally Pesin’s formula is \(\sum _{q_0,q_1,\ldots ,q_N} e^{-\lambda _u(S^jx)}=O(1)\), see Eq. (3.8.6) and, formally, \(s(\mu _{srb})-\mu _{srb}(\lambda _u)=0\), where \(s(\mu )\) is the Kolmogorov-Sinai entropy, see p. 73, and \(\mu (\lambda )\mathop {=}\limits ^{def}\int \lambda \,d\mu \). Furthermore \(s(\mu )-\mu (\lambda _u)\) is maximal at \(\mu =\mu _{srb}\): “Ruelle’s variational principle”. See p. 60 and [10, Proposition 6.3.4].
- 13.
I would say a strong one.
- 14.
In equilibrium, under the ergodic hypothesis, all microcells are non transient and the SRB distribution coincides with the Liouville distribution.
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Gallavotti, G. (2014). Discrete Phase Space. In: Nonequilibrium and Irreversibility. Theoretical and Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-06758-2_3
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