Abstract
The idea that natural phenomena proceed in a well-defined temporal direction, and therefore that the past is clearly distinguishable from the future, is based on indisputable empirical evidence.
Nothing is more practical than a good theory.
L. Boltzmann.
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Notes
- 1.
Which means that the dynamics remain bounded both in position and velocity.
- 2.
Which is described by a probability density \( \rho _0 ({\mathbf x}) \) localised around a certain point \({\mathbf x}^c{:}\) this can indeed reflect our imprecise knowledge of the initial state of the system.
- 3.
As a matter of fact, the opposite assumption is equally viable dynamically, and leads to the so-called anti-Boltzmann equation.
- 4.
At any finite \(N,\) the evolution is reversible. Therefore, from a conceptual, qualitative point of view, the case with \(0<\frac{1}{N}\ll 1\) differs from that with \(\frac{1}{N}=0.\)
- 5.
For instance, from the point of view of the pores in the marble, all cathedrals look the same, and yet they all look different from a certain distance.
- 6.
At that time the existence of atoms was still an open problem.
- 7.
For instance the statistical features of a colloidal particle, which is much bigger than the single molecules, is described by a Fokker-Planck equation which takes into account a possible external potential and the interactions between the colloidal particle and the molecules.
- 8.
Hilbert’s ideas had been further developed by Chapman and Enskog who introduced an efficient, although not fully rigorous method of computing transport coefficients (such as the viscosity) in terms of the microscopic interactions among the molecules(Chapman and Cowling 1970; Cercignani et al. 1994).
- 9.
We write just the systematic deterministic part. The complete Langevin equation also contains a noise term (as in Sect. 3.3).
- 10.
In same cases, as in the presence of complicated spatial geometry, the cellular automata approach has practical advantages over the usual standard numerical methods.
- 11.
During the first world war, when he was an ambulance driver on the French front, Richardson wrote the first draft of his book on numerical weather forecasting. His manuscript, lost during the battle of Champagne (April 1917), was fortunately recovered under a heap of coal, several months later. Besides meteorology, Richardson wrote seminal works to numerical analysis, turbulence, modelling in psychology and to the discovery of some aspects of fractals. The first to pose the question about the length of the coast of Britain was not Mandelbrot, as commonly believed, but Richardson.
- 12.
In the numerical integration of partial differential equations, one has to discretise space, with a grid of given size \(\varDelta x,\) and time, with an integration step \(\varDelta t.\) The numerical results then converge to the solution of the partial differential equation only if the ratio \(\varDelta t /\varDelta x\) is smaller than a number \(C,\) which depends on the equation and, typically, also on the initial conditions. This bound is called the “Courant condition” (Courant et al. 1928; Press et al. 1986). As an example, consider the following partial differential equation:
$$\begin{aligned} {{\partial u(x,t)} \over {\partial t}} =-v {{\partial u(x,t)} \over {\partial x}} \end{aligned}$$where \( v\) is a constant and \(-\infty < x < \infty .\) A simple finite-difference approximation of this equation yields
$$\begin{aligned} {{u_j^{n+1}-u_j^{n}} \over {\varDelta t}}=-v {{u_{j+1}^{n}-u_{j-1}^{n}} \over {2 \varDelta x}} \end{aligned}$$where \(u_j^n\) is the value taken by \(u\) in the point \(j \varDelta x\) at time \(n {\varDelta }t{:}u_j^n=u(j \varDelta x, n \varDelta t).\) In this case the Courant condition for the stability of the algorithm reads
$$\begin{aligned} {\varDelta t \over \varDelta x} \le C= { 1 \over |v|}. \end{aligned}$$Here, \(C\) does not depend on the initial condition, because the equation is linear.
- 13.
The Meteorological Project, developed at the Institute for Advanced Study, in Princeton, involved scientists from different fields, including leading mathematicians such as J. von Neumann, experts of meteorology, engineers and computer programmers. This project began to realise Richardson’s dream. Three fundamental issues were tackled by the project: technology, which led to the design of the first modern computer, ENIAC; numerical methods for the integration of partial differential equations; and the introduction of effective equations for meteorology.
- 14.
A simple example which illustrates how an effective equation for large scale behaviour can be obtained is the following, (Frisch 1995; E and Engquist 2003): consider the diffusion equation in one spatial dimension:
$$\begin{aligned} { \partial \over {\partial t}} \theta = { \partial \over {\partial x}}\left( D(x, {x / \varepsilon }) { \partial \theta \over {\partial x}}\right) \end{aligned}$$where the coefficient \(D(x, {x \over \varepsilon })\) contains two scales: a scale of \(O(\varepsilon )\) and a scale of \(O(1).\) For instance, \(D(x,y)\) could be periodic in \(y\) with period 1. The above system describes physical processes such as heat conduction in a composite material. The aim is to write an effective diffusion equation valid at long times and scales much larger than \(\varepsilon ,\) i.e. an equation of the form:
$$\begin{aligned} { \partial \over {\partial t}} \theta = { \partial \over {\partial x}}\Bigr (D^E(x) { \partial \over {\partial x}}\theta \Bigl ), \end{aligned}$$where \(D^E(x)\) must be obtained in terms of \(D(x,y).\) The result is simply given by the harmonic average, computed over the variable \(y\) and expressed by:
$$\begin{aligned} D^E(x)= {1 \over \left\langle {1 \over D(x,y)} \right\rangle }_y = \left[ \int \limits _0^1 {dy \over D(x,y)} \right] ^{-1}{.} \end{aligned}$$ - 15.
These equations are called quasigeostrophic; their simplest instance is a barotropic equation, in which the pressure depends only on the horizontal location.
- 16.
This equation had already been used by Rossby to study atmospheric waves, but before the results of Charney and coworkers, nobody seriously believed that the model could have produced quantitatively accurate predictions.
- 17.
and surely our point of view is shared by many physicists.
- 18.
For example, Hoover (1999) claims that Our exploration of time reversibility from the perspective of computer simulation and chaos has provided us with insights into the breaking of the time symmetry which were not available to Boltzmann or Gibbs [...] Simulations have clarified the formation and significance of time-reversible ergodic multifractal phase-space structures.
References
Barnum, H., Caves, C.M., Fuchs, C., Schack, R., Driebe, D.J., Hoover, W.G., Posch, H., Holian, B.L., Peierls, R., Lebowitz, J.L.: Is Boltzmann entropy time’s arrow’s archer? Phys. Today 47, 11 (1994)
Boltzmann, L.: Theoretical Physics and Philosophical Problems: Selected Writings. Springer, New York (1974)
Bricmont, J.: Science of chaos or chaos in science? Ann. N.Y. Acad. Sci. 775(1), 131 (1995)
Cecconi, F., Cencini, M., Vulpiani, A.: Transport properties of chaotic and non-chaotic many particle systems. J. Stat. Mech. Theory Exp. P12001 (2007)
Cencini, M., Cecconi, F., Vulpiani, A.: Chaos: From Simple Models to Complex Systems. World Scientific, Singapore (2009)
Cercignani, C., Illner, R., Pulvirenti, M.: The Mathematical Theory of Dilute Gases. Springer, New York (1994)
Cercignani, C.: Ludwig Boltzmann: The Man Who Trusted Atoms, vol. Oxford. Oxford University Press, USA (2006)
Chapman, S., Cowling, T.G.: The mathematical theory of non-uniform gases: an account of the kinetic theory of viscosity, thermal conduction and diffusion in gases. Cambridge University Press, Cambridge (1970)
Courant, R., Friedrichs, K., Lewy, H.: Uber die partiellen differenzengleichungen der mathematischen physik. Math. Ann. 100(1), 32 (1928)
Cronin, J.W.: CP symmetry violation the search for its origin. Rev. Mod. Phys. 53(3), 373 (1981)
Duhem, P.: L’évolution de la mécanique. Revue générale des sciences pures et appliques XIV (1903)
Ehrenfest, P., Ehrenfest, T.: The Conceptual Foundations of the Statistical Approach in Mechanics. Dover Publications, New York (2002)
E, W., Engquist, B.: Multiscale modeling and computation. Not. AMS 50, 1062 (2003)
Frisch, U., Hasslacher, B., Pomeau, Y.: Lattice-gas automata for the Navier-Stokes equation. Phys. Rev. Lett. 56(14), 1505 (1986)
Frisch, U.: Turbulence. Cambridge University Press, Cambridge (1995)
Gorban, A.N., Karlin, I.V., Zinovyev, A.Y.: Constructive methods of invariant manifolds for kinetic problems. Phys. Rep. 396(4), 197 (2004)
Grad, H.: On the kinetic theory of rarefied gases. Commun. Pure Appl. Math. 2(4), 331 (1949)
Guerra, F.: Reversibilità/irreversibilità. Enciclopedia Einaudi Torino XI, 1067 (1980)
Haurwitz, B.: Dynamic Meteorology. McGraw-Hill, New York (1941)
Hoover, W.G.: Time Reversibility, Computer Simulation, and Chaos. World Scientific, Singapore (1999)
Kac, M.: Probability and Related Topics in Physical Sciences. American Mathematical Society, Providence (1957)
Lanford, O.E.: The hard sphere gas in the Boltzmann-Grad limit. Phys. A: Stat. Theoret. Phys. 106(1–2), 70 (1981)
Lebowitz, J.L.: Boltzmann’s entropy and time’s arrow. Phys. Today 46, 32 (1993)
Lieb, E.H., Yngvason, J.: A fresh look at entropy and the second law of thermodynamics. Phys. Today 53 (2000)
Nicolis, G., Prigogine, I.: Self-organization in Nonequilibrium Systems. Wiley, New York (1977)
Orban, J., Bellemans, A.: Velocity-inversion and irreversibility in a dilute gas of hard disks. Phys. Lett. A 24, 620 (1967)
Poincaré, H.: Sur le problme des trois corps et les quations de la dynamique. Acta Math. 13, 1 (1890)
Poincaré, H.: Le mecanisme et l’experience. Revue de Métaphysique et de Morale 1(6), 534 (1893)
Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flanner, B.P.: Numerical Recipes: The Art of Scientific Computing. Cambridge University Press, Cambridge (1986)
Prigogine, I., Stengers, I.: La Nouvelle Alliance. Gallimard, Paris (1979)
Prigogine, I., Stengers, I.: Order out of Chaos. Bantam Books, Toronto (1984)
Prigogine, I.: Les lois du chaos. Flammarion, Paris (1994)
Primas, H.: Emergence in exact natural science. Acta Polytech. Scand. 91, 83 (1998)
Richardson, L.F.: Weather Prediction by Numerical Methods. Cambridge University Press, Cambridge (1922)
Rothman, D.H., Zaleski, S.: Lattice-gas cellular automata: simple models of complex hydrodynamics. Cambridge University Press, Cambridge (2004)
Saint-Raymond, L.: Hydrodynamic limits of the Boltzmann equation. Springer, New York (2009)
Villani, C.: A review of mathematical topics in collisional kinetic theory. Handb. Math. Fluid Dyn. 1, 71 (2002)
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Chibbaro, S., Rondoni, L., Vulpiani, A. (2014). From Microscopic Reversibility to Macroscopic Irreversibility. In: Reductionism, Emergence and Levels of Reality. Springer, Cham. https://doi.org/10.1007/978-3-319-06361-4_4
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