Non-equilibrium Statistical Mechanics of a Driven Lattice Gas Model: Probability Function, FDT-violation, and Monte Carlo Simulations
The study of non-equilibrium systems has attracted
increasing interest in recent years, mainly due to the lack of
theoretical frameworks, unlike their equilibrium counterparts.
Studying the steady state and/or simple systems is thus one of the
main interests. Hence in this work we have focused our attention on
the driven lattice gas model (DLG model) consisting of interacting
particles subject to an external field E. The dynamics of the system
are given by hopping of particles to nearby empty sites with rates
biased for jumps in the direction of E. Having used small two
dimensional systems of DLG model, the stochastic properties at nonequilibrium
steady state were analytically studied. To understand the
non-equilibrium phenomena, we have applied the analytic approach
via master equation to calculate probability function and analyze
violation of detailed balance in term of the fluctuation-dissipation
theorem. Monte Carlo simulations have been performed to validate
the analytic results.
[1] J. W. Dufty and J. Lutsko, Recent Developments in Non-equilibrium
Thermodynamics, J. Casas-Vazques, D. Jou and J. M. Rubi, Ed. Berlin:
Springer, 1986.
[2] J. Krug and H. Spohn, Solids Far from Equilibrium: Growth, Morphology
and Defects, C. Godreche, Ed. New York: Cambridge University Press,
1991.
[3] G. Nicolis and I. Prigogine, Self-Organization in Non-equilibrium Systems.
New York: Wiley, 1977.
[4] R. Bhattacharya and M. Majumdar,,Random dynamical systems and
chaos: Theory and Application, Cambridge: Cambridge University Press,
2007, pp.119-239.
[5] S. Perrett, S. J. Freeman, P. J. G. Butler and A. R., "Equilibrium folding
properties of the yeast prion protein determinant Ure2", J. Mol. Biol., vol.
290, no. 1, pp. 331-345, Jul. 1999.
[6] J. Wong-ekkabut, W. Triampo, I.-Ming Tang, D. Triampo, D. Baowan,
and Y. Lenbury, "Vacancy-Mediated Disordering Process in Binary
Alloys at Finite Temperatures: Monte Carlo Simulations", J. Korean
Phys. Soc., vol. 45, 2003.
[7] Bar-Yam Yaneer, Dynamical of Complex Systems, Boston: Addison-
Wesley,1997,.pp.38-57.
[8] E. Ben-Naim, H. Frauenfelder, Z. Toroczkai, Complex Networks, Berlin:
Springer, 2004,. pp. 51-88
[9] R. Ash, Information theory, New York: Wiley, 1965, pp.169-210.
[10] J. Avery, Information Theory and Evolution, Toh Tuck Lin: World
Scientific, 2003
[11] Econophysics and Sociophysics, B. K. Chakrabarti, A. Chakraborti, and
A. Chatterjee, Ed. New York: WILEY-VCH, 2006,.pp. 65-88.
[12] P. C. Martin and J. Schwinger, "Theory of many-particle systems. I".
Phys. Rev., vol. 115, no. 6, pp. 1342 - 1373, Mar. 1959.
[13] J.-P. Eckmann and I. Procaccia, "Onset of defect-mediated turbulence".
Phys. Rev. Lett., vol. 66, no. 7, pp. 891-894, May. 1991.
[14] J. Casas-Vázquez and D. Jou, "Nonequilibrium temperature versus localequilibrium
temperature", Phys. Rev. E, vol. 49, no. 2, pp. 1040 - 1048,
Aug. 1994.
[15] A. Campos and BL. Hu, "Nonequilibrium dynamics of a thermal plasma
in a gravitational field". Phys. Rev. D, vol. 58, no. 12, pp. 125021, Feb.
1998.
[16] D. Boyanovsky, F. Cooper, H. J. de Vega, and P. Sodano, "Evolution of
inhomogeneous condensates: Self-consistent variational approach". Phys.
Rev. D, vol. 58, pp. 025007, Feb. 1998.
[17] G. Torrieri, S. Jeon, and J. Rafelski, "Particle yield fluctuations and
chemical nonequilibrium in Au-Au collisions at sqrt [s_ {NN}]= 200
GeV", Phys. Rev. C, vol. 74, no. 2, pp. 024901, Feb. 2006.
[18] W. Kwak, Y. Jae-Suk, K. In-mook, and D. P. Landau, "Sub-block order
parameter in a driven Ising lattice gas using block distribution functions",
Phys. Rev. E, vol. 75, no. 4, pp. 041108, Nov. 2007.
[19] J. Honerkamp, Stochastic Dynamical Systems: Concepts, Numerical
Methods, Data Analysis. New York: VCH. , 1994.
[20] M. Blume, V. J. Emery, and Robert B. Griffiths, "Ising Model for the
Transition and Phase Separation in He^{3}-He^{4} Mixtures", Phys. Rev.
A, vol. 4, no. 3, pp. 1071-1077, Mar. 1971.
[21] P. C. Hohenberg and B. I. Halperin, "Theory of dynamic critical
phenomena", Rev. Mod. Phys., vol. 49, no. 3, pp. 435-479, Jul. 1977.
[22] K. G. Wilson, "The renormalization group: Critical phenomena and the
Kondo problem", Rev. Mod. Phys., vol. 47, no. 4, pp. 773-840, Jul. 1975.
[23] B. Schmittmann and R. K. P. Zia, Statistical Mechanics of Driven
Diffusive Systems , C. Domb and J. L. Lebowitz, Ed. New York:
Academic Press, 1995.
[24] S. Katz, J. Lebowitz, H. Spohn, "Phase transitions in stationary
nonequilibrium states of model lattice systems", Phys. Rev. B, vol. 28, no.
3, pp. 1655-1658, Dec. 1983.
[25] K. G. Wilson, "The renormalization group and critical phenomena ",
Rev. Mod. Phys., vol. 55, no. 3, pp.583-600, Jul. 1983.
[26] R. R. Netz and A. Aharony, "Critical behavior of energy-energy, strainstrain,
higher-harmonics, and similar correlation functions", Phys. Rev. E,
vol. 55, no. 3, pp. 2267-2278, Aug. 1997.
[27] Y. Chen, "Short-time critical behavior of anisotropic cubic systems",
Phys. Rev. B, vol. 63, no. 9, pp. 092301, Sep. 2001.
[28] J. Marro, J. L. Vallés, and J. M. Gonz├ílez-Miranda, "Critical behavior in
nonequilibrium phase transitions", Phys. Rev. B, vol. 35, no. 7, pp. 3372-
3375, Jul. 1987.
[29] I. Svare, F. Borsa, D. R. Torgeson, and S. W. Martin, "Correlation
functions for ionic motion from NMR relaxation and electrical
conductivity in the glassy fast-ion conductor (Li_ {2} S) _ {0.56}(SiS_
{2}) _ {0.44}", Phys. Rev. B, vol. 48, no. 13, pp. 9336-9344, Mar. 1993.
[30] S. Adams and J. Swenson, "Determining ionic conductivity from
structural models of fast ionic conductors", Phys. Rev. Lett., vol. 84, no.
18, pp. 4144-4147, Sep. 2000.
[31] N. Metropolis, A.W. Rosenbluth, M.M. Rosenbluth, A.H. Teller and E.
Teller, "Equation of state calculations by fast computing machines", J.
Chem. Phys., vol. 21, no. 6, pp.1087, Mar. 1953.
[32] F. Spitzer, "Interaction of Markov processes", Adv. Math., vol. 5, no. 2,
pp. 246-290, 1970.
[33] R.K.P. Zia and T. Blum, Scale Invariance, Interfaces, and Nonequilibrium
Dynamics, A. Mckane et al., Ed. New York: Plenum Press,
1995.
[34] W. Triampo, I.M. Tang, J. Wong-Ekkabut, "Explicit Calculations on
Small Non-Equilibrium Driven Lattice Gas Models", J. Korean Phys.
Soc., vol. 43, no. 2, pp. 207-214, 2003.
[35] L. Onsager, " Crystal statistics. I. A two-dimensional model with an
order-disorder transition", Phys. Rev., vol. 65, no. 3-4, pp.117-149, Oct.
1944.
[36] L.P. Kadanoff, "Critical Phenomena", Proceedings of International
School of Physics "Enrico Fermi,", Course 51, M. S. Green, Ed. New
York: Academic, 1971.
[1] J. W. Dufty and J. Lutsko, Recent Developments in Non-equilibrium
Thermodynamics, J. Casas-Vazques, D. Jou and J. M. Rubi, Ed. Berlin:
Springer, 1986.
[2] J. Krug and H. Spohn, Solids Far from Equilibrium: Growth, Morphology
and Defects, C. Godreche, Ed. New York: Cambridge University Press,
1991.
[3] G. Nicolis and I. Prigogine, Self-Organization in Non-equilibrium Systems.
New York: Wiley, 1977.
[4] R. Bhattacharya and M. Majumdar,,Random dynamical systems and
chaos: Theory and Application, Cambridge: Cambridge University Press,
2007, pp.119-239.
[5] S. Perrett, S. J. Freeman, P. J. G. Butler and A. R., "Equilibrium folding
properties of the yeast prion protein determinant Ure2", J. Mol. Biol., vol.
290, no. 1, pp. 331-345, Jul. 1999.
[6] J. Wong-ekkabut, W. Triampo, I.-Ming Tang, D. Triampo, D. Baowan,
and Y. Lenbury, "Vacancy-Mediated Disordering Process in Binary
Alloys at Finite Temperatures: Monte Carlo Simulations", J. Korean
Phys. Soc., vol. 45, 2003.
[7] Bar-Yam Yaneer, Dynamical of Complex Systems, Boston: Addison-
Wesley,1997,.pp.38-57.
[8] E. Ben-Naim, H. Frauenfelder, Z. Toroczkai, Complex Networks, Berlin:
Springer, 2004,. pp. 51-88
[9] R. Ash, Information theory, New York: Wiley, 1965, pp.169-210.
[10] J. Avery, Information Theory and Evolution, Toh Tuck Lin: World
Scientific, 2003
[11] Econophysics and Sociophysics, B. K. Chakrabarti, A. Chakraborti, and
A. Chatterjee, Ed. New York: WILEY-VCH, 2006,.pp. 65-88.
[12] P. C. Martin and J. Schwinger, "Theory of many-particle systems. I".
Phys. Rev., vol. 115, no. 6, pp. 1342 - 1373, Mar. 1959.
[13] J.-P. Eckmann and I. Procaccia, "Onset of defect-mediated turbulence".
Phys. Rev. Lett., vol. 66, no. 7, pp. 891-894, May. 1991.
[14] J. Casas-Vázquez and D. Jou, "Nonequilibrium temperature versus localequilibrium
temperature", Phys. Rev. E, vol. 49, no. 2, pp. 1040 - 1048,
Aug. 1994.
[15] A. Campos and BL. Hu, "Nonequilibrium dynamics of a thermal plasma
in a gravitational field". Phys. Rev. D, vol. 58, no. 12, pp. 125021, Feb.
1998.
[16] D. Boyanovsky, F. Cooper, H. J. de Vega, and P. Sodano, "Evolution of
inhomogeneous condensates: Self-consistent variational approach". Phys.
Rev. D, vol. 58, pp. 025007, Feb. 1998.
[17] G. Torrieri, S. Jeon, and J. Rafelski, "Particle yield fluctuations and
chemical nonequilibrium in Au-Au collisions at sqrt [s_ {NN}]= 200
GeV", Phys. Rev. C, vol. 74, no. 2, pp. 024901, Feb. 2006.
[18] W. Kwak, Y. Jae-Suk, K. In-mook, and D. P. Landau, "Sub-block order
parameter in a driven Ising lattice gas using block distribution functions",
Phys. Rev. E, vol. 75, no. 4, pp. 041108, Nov. 2007.
[19] J. Honerkamp, Stochastic Dynamical Systems: Concepts, Numerical
Methods, Data Analysis. New York: VCH. , 1994.
[20] M. Blume, V. J. Emery, and Robert B. Griffiths, "Ising Model for the
Transition and Phase Separation in He^{3}-He^{4} Mixtures", Phys. Rev.
A, vol. 4, no. 3, pp. 1071-1077, Mar. 1971.
[21] P. C. Hohenberg and B. I. Halperin, "Theory of dynamic critical
phenomena", Rev. Mod. Phys., vol. 49, no. 3, pp. 435-479, Jul. 1977.
[22] K. G. Wilson, "The renormalization group: Critical phenomena and the
Kondo problem", Rev. Mod. Phys., vol. 47, no. 4, pp. 773-840, Jul. 1975.
[23] B. Schmittmann and R. K. P. Zia, Statistical Mechanics of Driven
Diffusive Systems , C. Domb and J. L. Lebowitz, Ed. New York:
Academic Press, 1995.
[24] S. Katz, J. Lebowitz, H. Spohn, "Phase transitions in stationary
nonequilibrium states of model lattice systems", Phys. Rev. B, vol. 28, no.
3, pp. 1655-1658, Dec. 1983.
[25] K. G. Wilson, "The renormalization group and critical phenomena ",
Rev. Mod. Phys., vol. 55, no. 3, pp.583-600, Jul. 1983.
[26] R. R. Netz and A. Aharony, "Critical behavior of energy-energy, strainstrain,
higher-harmonics, and similar correlation functions", Phys. Rev. E,
vol. 55, no. 3, pp. 2267-2278, Aug. 1997.
[27] Y. Chen, "Short-time critical behavior of anisotropic cubic systems",
Phys. Rev. B, vol. 63, no. 9, pp. 092301, Sep. 2001.
[28] J. Marro, J. L. Vallés, and J. M. Gonz├ílez-Miranda, "Critical behavior in
nonequilibrium phase transitions", Phys. Rev. B, vol. 35, no. 7, pp. 3372-
3375, Jul. 1987.
[29] I. Svare, F. Borsa, D. R. Torgeson, and S. W. Martin, "Correlation
functions for ionic motion from NMR relaxation and electrical
conductivity in the glassy fast-ion conductor (Li_ {2} S) _ {0.56}(SiS_
{2}) _ {0.44}", Phys. Rev. B, vol. 48, no. 13, pp. 9336-9344, Mar. 1993.
[30] S. Adams and J. Swenson, "Determining ionic conductivity from
structural models of fast ionic conductors", Phys. Rev. Lett., vol. 84, no.
18, pp. 4144-4147, Sep. 2000.
[31] N. Metropolis, A.W. Rosenbluth, M.M. Rosenbluth, A.H. Teller and E.
Teller, "Equation of state calculations by fast computing machines", J.
Chem. Phys., vol. 21, no. 6, pp.1087, Mar. 1953.
[32] F. Spitzer, "Interaction of Markov processes", Adv. Math., vol. 5, no. 2,
pp. 246-290, 1970.
[33] R.K.P. Zia and T. Blum, Scale Invariance, Interfaces, and Nonequilibrium
Dynamics, A. Mckane et al., Ed. New York: Plenum Press,
1995.
[34] W. Triampo, I.M. Tang, J. Wong-Ekkabut, "Explicit Calculations on
Small Non-Equilibrium Driven Lattice Gas Models", J. Korean Phys.
Soc., vol. 43, no. 2, pp. 207-214, 2003.
[35] L. Onsager, " Crystal statistics. I. A two-dimensional model with an
order-disorder transition", Phys. Rev., vol. 65, no. 3-4, pp.117-149, Oct.
1944.
[36] L.P. Kadanoff, "Critical Phenomena", Proceedings of International
School of Physics "Enrico Fermi,", Course 51, M. S. Green, Ed. New
York: Academic, 1971.
@article{"International Journal of Chemical, Materials and Biomolecular Sciences:58028", author = "K. Sudprasert and M. Precharattana and N. Nuttavut and D. Triampo and B. Pattanasiri and Y. Lenbury and W. Triampo", title = "Non-equilibrium Statistical Mechanics of a Driven Lattice Gas Model: Probability Function, FDT-violation, and Monte Carlo Simulations", abstract = "The study of non-equilibrium systems has attracted
increasing interest in recent years, mainly due to the lack of
theoretical frameworks, unlike their equilibrium counterparts.
Studying the steady state and/or simple systems is thus one of the
main interests. Hence in this work we have focused our attention on
the driven lattice gas model (DLG model) consisting of interacting
particles subject to an external field E. The dynamics of the system
are given by hopping of particles to nearby empty sites with rates
biased for jumps in the direction of E. Having used small two
dimensional systems of DLG model, the stochastic properties at nonequilibrium
steady state were analytically studied. To understand the
non-equilibrium phenomena, we have applied the analytic approach
via master equation to calculate probability function and analyze
violation of detailed balance in term of the fluctuation-dissipation
theorem. Monte Carlo simulations have been performed to validate
the analytic results.", keywords = "Non-equilibrium, lattice gas, stochastic process", volume = "5", number = "6", pages = "487-9", }
