Abstract
Nonequilibrium stationary states are studied for a multibaker map, a simple reversible chaotic dynamical system. The probabilistic description is extended by representing a dynamical state in terms of a measure instead of a density function. The equation of motion for the cumulative function of this measure is derived and stationary solutions are constructed with the aid of deRham-type functional equations. To select the physical states, the time evolution of the distribution under a fixed boundary condition is investigated for an open multibaker chain of scattering type. This system corresponds to a diffusive flow experiment through a slab of material. For long times, any initial distribution approaches the stationary one obeying Fick's law. At stationarity, the intracell distribution is singular in the stable direction and expressed by the Takagi function, which is continuous but has no finite derivatives. The result suggests that singular measures play an important role in the dynamical description of non-equilibrium states.
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References
J. Lebowitz and H. Spohn,J. Stat. Phys. 19:633 (1978); H. Spohn,Rev. Mod. Phys. 53:569 (1980); L. A. Bunimovich and Y. G. Sinai,Commun. Math. Phys. 78:479 (1980).
S. Grossman and H. Fujisaka,Phys. Rev. A 26:1779 (1982).
S. Thomae, Chaos-induced diffusion, inStatics and Dynamics of Nonlinear Systems, G. Benedeket al. eds. (Springer, Berlin, 1983), pp. 204–210.
P. Gaspard,Phys. Lett. A 168:13 (1992).
H. H. Hasegawa and D. J. Driebe,Phys. Lett. A 168:18 (1992).
P. Gaspard,J. Stat. Phys. 68:673 (1992).
P. Gaspard,Chaos 3:427 (1993).
H. H. Hasegawa and D. J. Driebe,Phys. Rev. E 50:1781 (1994).
S. Tasaki, A. Hakmi, and I. Antoniou, To appear (1994).
M. Pollicott,Invent. Math. 81:413 (1985);85:147 (1986).
D. Ruelle,Phys. Rev. Lett. 56:405 (1986);J. Stat. Phys. 44:281 (1986).
P. Gaspard and S. Tasaki, To appear.
I. Prigogine,Non-equilibrium Statistical Mechanics (Wiley, New York, 1962).
I. Prigogine, C. George, F. Henin, and L. Rosenfeld,Chem. Scripta 4:5–32 (1973);R. Balescu,Equilibrium and Nonequilibrium Statistical Mechanics (Wiley, New York, 1975); T. Petrosky and I. Prigogine,Physica A 175:146–209 (1991);Proc Natl. Acad. Sci. USA 90:9393 (1993); I. Antoniou and I. Prigogine,Physica A 192:443–464 (1993).
J.-P. Eckmann and D. Ruelle,Rev. Mod. Phys. 57:617 (1985).
D. Ruelle,Chaotic Evolution and Strange Attractors (Cambridge University Press, Cambridge, 1989).
S. Tasaki and P. Gaspard, InTowards the Harnessing of Chaos, M. Yamaguti, ed. (Elsevier, Amsterdam, 1994), pp. 273–288.
R. C. Toleman,The Principles of Statistical Mechanics (Oxford University Press, Oxford, 1938; reprinted Dover, New York, 1979).
G. deRham,Rend. Sem. Mat. Torino 16: 101 (1957).
M. Hata and M. Yamaguti,Jpn. J. Appl. Math. 1:183 (1984).
M. Hata, InPatterns and Waves, T. Nishida, M. Mimura, and H. Fujii, eds. (Kinokuniya, Tokyo, and North-Holland, Amsterdam, 1986), pp. 259–278.
S. Tasaki, I. Antoniou, and Z. Suchanecki,Phys. Lett. A 179:97 (1993).
E. Hewitt and K. Stromberg,Real and Abstract Analysis (Springer-Verlag, New York, 1975).
P. Billingsley,Ergodic Theory and Information (Wiley, New York, 1965).
T. Takagi,Proc. Phys. Math. Soc. Japan Ser. II 1:176 (1903).
D. Ornstein,Science 243:182 (1989).
I. P. Cornfeld and Ya. G. Sinai, InDynamical Systems II (Springer-Verlag, Berlin, 1989).
Y. Oono and Y. Takahashi,Prog. Theor. Phys. 63:1804 (1980); Y. Takahashi and Y. Oono,Prog. Theor. Phys. 71:851 (1984).
S. Tasaki, Z. Suchanecki, and I. Antoniou,Phys. Lett. A. 179:103 (1993).
T. Tél and E. Ott, eds., Chaotic Scattering,Chaos 3(4):417–706 (1993).
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Tasaki, S., Gaspard, P. Fick's law and fractality of nonequilibrium stationary states in a reversible multibaker map. J Stat Phys 81, 935–987 (1995). https://doi.org/10.1007/BF02179299
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DOI: https://doi.org/10.1007/BF02179299