Kinetic equation for a gas with attractive forces as a functional equation

Ryszard Wojnar

Abstract


Diffusion problems studied in the time scale comparable with time of particles collision lead to kinetic equations which for step-wise potentials are functional equations in the velocity space. After a description of meaning of diffusion in biology and survey of derivation of kinetic equations by projective operator method, we pay an attention to the Lorentz gas with step potential. The gas is composed of N particles: N-1 of which are immovable between N-1 immovable particles-scatterers, particle number 1 is moving, and we describe its movement by means of one-particle distribution function satisfying a kinetic equation. Solutions of the kinetic equation for some simple potentials are given. We derive also a kinetic equation for one-dimensional Lorentz gas, which is a functional equation.

Keywords


kinetic equation; distribution function; functional equation

Mathematics Subject Classification


82C41, 82C70, 92B05

References


Agutter, P.S., P. Colm Malone and D.N. Wheatley. "Intracellular transport Mechanisms: A critique of diffusion theory." J. Theoret. Biol. 176 (1995), 261-272.

---. "Diffusion theory in biology: A relic of mechanistic materialism." J. Histol. Biol. 33 (2000): 71-111.

Astumian, R.D. and P. Hanggi. "Brownian motors." Phys. Today 55 (2002): 33-39.

Beijeren, H., van and O. Mülken. "Thermodynamic formalism for the Lorentz gas with open boundaries in d dimensions." Phys. Rev. E 71 (2005): 036213.

Bigwood, E.J. "The distribution of ions in gels." Trans. Faraday Soc. 26 (1930): 704-719.

Bobylev, A.V. et al. "There is more to be learned from the Lorentz model." J. Statist. Phys. 87 (1997): 1205-1228.

Boley, Ch.D. "Projection-operator approach to a renormalized kinetic theory." Phys. Rev. A 11 (1975): 328-339.

Brown, R. "A brief account of microscopical observation made in the months of June, July and August, 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies." Privately printed pamphlet, 1828.

Bustamante, C. et al. "Mechanical processes in biochemistry." Ann. Rev. Biochem. 73 (2004): 705-748.

"Cartilage." Wikipedia. Wikimedia Foundation, Inc. http://en.wikipedia.org/wiki/Cartilage.

Chapman, S. and T.G. Cowling. "The mathematical theory of non-uniform gases." Cambridge: 1939.

Darwin, C.G. and R.H. Fowler. "On the partition of energy." Philos. Mag. 44 (1922): 450-479.

Dasgupta, R., T.K. Ballabh and S. Tarafdar. "Quasielastic incoherent scattering in fractal systems." Phys. Rev. B 52 (1995): 10021-10024.

Deng, Yi, J. Bechhoefer and N.R. Forde. "Brownian motion in a modulated optical trap." J. Opt. A: Pure Appl. Opt. 9 (2007): S256-S263.

Egelstaff, P.A. "Thermal neutron scattering." New York: Academic Press, 1965.

Einstein, A. "Von der molekulärkinetischen Theorie der Wärme gefordete Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen." Ann. Phys. 17 (1905): 549-554.

Ernst, M.H. and P.M. Binder. "Lorentz lattice gases: basic theory." J. Statist. Phys. 51 (1988): 981-990.

Fermi, E., J. Pasta and S. Ulam. "Studies of Nonlinear Problems." 1955. Document LA-1940.

Fick, A.E. "Über Diffusion." Annalen der Physik und Chemie von J.C. Poggendorff 94 (1855): 59-86.

---. "Gesammelte Schriften." Würzburg: Bd.I, 1904.

---. "Medizinische Physik." Zürich: Kapelmann, 1856.

Ford, B.J. "Brownian movement in Clarkia pollen: A reprise of the first observations." Microscope 40 (1992): 235-241.

Gilson, M.K. et al. "The statisticalthermodynamic basis for computation of binding afinities: a critical review." Biophys. J. 72 (1997): 1047-1069.

Gilson, M.K. and H.-X. Zhou. "Calculation of protein-ligand binding afinities." Annual Review of Biophysics and Biomolecular Structure 36 (2007): 21-42.

Goldston, R.J. and P.H. Rutherford. "Introduction to plasma physics." New York: CRC Press, 1995.

González-Candela, E. and V. Romero-Rochín. "Overdamped thermal ratchets in one and more dimensions. Kinesin transport and protein folding." Physica A: Statistical Mechanics and its Applications 372 (2006): 249-262.

Harayama, T., R. Klages and P. Gaspard. "Deterministic diffusion in flower-shaped billiards." Phys. Rev. E 66 (2002): 026211.

Hauge, E.H. "Exact and Chapman-Enskog solutions of the Boltzmann equation for the Lorentz model." Phys. Fluids 13 (1970): 1201-1208.

---. "What can one learn from Lorentz models: Transport Phenomena." Lecture Notes in Phys. 31. Berlin: Springer, 1974. 337-367.

Hochachka, P.W. "The metabolic implications of intracellular circulation." Proc. Natl. Acad. Sci. USA 96 (1999): 12233-12239.

Höhler, G. "Quasielastic neutron scattering for the investigation of diffusive motions in solids and liquids." Berlin: Springer-Verlag, 1972.

"Homeostasis." Wikipedia. Wikimedia Foundation, Inc. http://en.wikipedia.org/wiki/Homeostasis.

Johnson, R.P.C. "A biologist sums up. In: Application of Laser Light to the Study of Biological Motion." New York: Plenum Press, 1983. 681-686.

Kac, M. "Random walk and the theory of Brownian motion." Amer. Math. Monthly 54 (1947): 369-391.

---. "Some stochastic problems in physics and mathematics." Lecture Pure Appl. Sci. 2. Magnolia Petroleum Company, 1956.

---. "Kilka zagadnień stochastycznych fizyki i matematyki." Warszawa: PWN, 1961.

---. "Statistical independence in probability, analysis and number theory." The Carus Mathematical Monographs 12. New York: Mathematical Association of America; John Wiley and Sons, Inc., 1959.

Kadanoff, L.P. and P.C. Martin. "Hydrodynamic equations and correlation functions." Ann. Physics 24 (1963): 419-469.

Klages, R. and Ch. Dellago. "Density-dependent diffusion in the periodic Lorentz gas." J. Statist. Phys. 101 (2000): 145-159.

Kuczma, M. "Functional equations in a single variable." Monographs Math. 46. Warszawa: PWN, 1968.

Lee, J.J. "Electron distribution function for a weakly ionized flowing gas in an electric field." Phys. Rev. Lett. 32 (1974): 1417-1420.

Lieb, E.H. ed. and D.C. Mattis ed. "Mathematical physics in one dimension: Exactly soluble models of interacting particles." Perspectives in Physics: a Series of Reprint Collections. New York: Academic Press, 1966.

Lindenfeld, M.J. "Self-structure factor of hard-sphere gases for arbitrary ratio of bath to test particle masses." J. Chem. Phys. 73 (1980): 5817-5829.

Longmuir, I.S. "Adaptation to Hypoxia." New York: Plenum Press, 1987.

---. "Adaptation to hypoxia." Advances in Experimental Medicine and Biology 21 (1987): 249-254.

Macheras, P. and A. Iliadis. "Modeling in biopharmaceutics, pharmacokinetics, and pharmacodynamics. Homogeneous and heterogeneous approaches." Interdisciplinary Applied Mathematics 30. New York: Springer, 2006.

Machta, J. and R. Zwanzig. "Diffusion in a periodic Lorentz gas." Phys. Rev. Lett. 50.25 (1983): 1959-1962.

Marklof, J. and A. Strömbergsson. "Kinetic transport in the two-dimensional periodic Lorentz gas." Nonlinearity 21 (2008): 1413-1422.

Mattis, D.C. "How to reduce practically any problem to one dimension." Springer Series in Solid State Sciences 23. Berlin and New York: Springer-Verlag, 1981.

Mattis D.C. ed. "The many-body problem - An encyclopedia of exactly solved models in one dimension." Singapore: World Scientific, 1994.

McKean, H.P. Jr. "Chapman-Enskog-Hilbert expansion for a class of solutions of the telegraph equation." J. Mathematical Phys. 8 (1967): 547-552.

Mityushev, V.V. and S.V. Rogosin. Constructive methods to linear and nonlinear boundary value problems of the analytic function. Theory and aplications." Monographs and Surveys in Pure Applied Mathematics 108. Boca Rota, Florida: Chapman&Hall/CRC, 2000.

Mori, H. "Transport, collective motion, and Brownian motion." Progr. Theoret. Phys. 33 (1965): 423-455.

Morita, T. and Y. Fukui. "Thermalization of the ideal gas in a one-dimensional box." Can. J. Phys. 57 (1979): 1103-1109.

Nakajima, S. "On quantum theory of transport phenomena: steady diffusion." Progr. Theoret. Phys. 20 (1958): 948-959.

Narbutowicz, M.A. "Kinetic equation for a heavy marked particle." Rep. Math. Phys. 8 (1975): 1-18.

Nelkin, M. and A. Ghatak. "Simple binary collision model for Van Hove's Gs(r; t)." Phys. Rev. (2) 135 (1964): A4-A9.

Peters, R. "Fluorescence microphotolysis to measure nucleocytoplasmic transport and intracellular mobility." Biochimica et Biophysica Acta 864 (1986): 305-359.

Peters, Sir, R.A. "Surface structure in cell activity." Trans. Faraday Soc. 26 (1930): 797-807.Prigogine, I. "Non-equilibrium statistical mechanics." New York: Interscience, 1964.

Reigada, R., A. Sarmiento and K . Lindenberg. "Breathers and thermal relaxation in Fermi-Pasta-Ulam arrays." arXiv:cond-mat/0210326v1.

Résibois, P. and M. De Leener. "Classical kinetic theory of fluids." New York, London, Sydney, Toronto: John Wiley&Sons, 1977.

Smith, D.A. and M.A. Geeves. "Strain-dependent cross-bridge cycle for muscle." Biophys. J. 69 (1995): 524-537.

Smith, D.A. "Direct tests of muscle cross-bridge theories: Predictions of a Brownian dumbbell model for position-dependent cross-bridge lifetimes and step sizes with an optically trapped Actin lament." Biophys. J. 75 (1998): 2996-3007.

Smoluchowski, M. von. "Zur kinetischen Theorie der Brownschen Molekulärbewegung und der Suspensionen." Ann. Phys. 21 (1906): 756-780.

Stecki, J. and H.S. Taylor. "On the areas of equivalence of the Bogoliubov theory and the Prigogine theory of irreversible processes in classical gases." Rev. Modern Phys. 37 (1965): 762-773.

Stecki, J. and R. Wojnar. "Kinetic equation for the self-correlation function G(r; t) in the dilute Boltzmann gas." Chem. Phys. Letters 2 (1968): 343-345.

Stecki, J. "Statistical thermodynamics." Warszawa: PWN, 1970.

---. "On the kinetic equation nonlocal in time for the generalized self-diffusion process." J. Computational Phys. 7 (1971): 547-553.

Streater, R.F. "Nonlinear heat equations." Rep. Math. Phys. 40 (1997): 557-564.

Swanson, J.M.J., R.H. Henchman and J.A. McCammon. "Revisiting free energy calculations: a theoretical connection to MM/PBSA and direct calculation of the association free energy." Biophys. J. 86 (2004): 67-74.

Szász, D. ed. "Hard ball systems and the Lorentz gas." New York: Springer, 2001.

Tsallis, C., E.P. Borges and F. Baldovin. "Mixing and equilibration: protagonists in the scene of nonextensive statistical mechanics." Phys. A 305.1-2 (2002): 1-18.

Ussing, H.H. "The distinction by means of tracers between active transport and diffusion." Acta Physiol. Scand. 19 (1949): 43-56.

---. "Transport of ions across cellular membranes." Physiol. Rev. 29 (1949): 127-155.

Vale, R.D. and F. Oosawa. "Protein motors and Maxwell's demons: Does mechanochemical transduction involve a thermal ratchet?." Adv. Biophys. 26 (1990): 97-134.

Van Hove, L. "Temperature variation of the magnetic inelastic scattering of slow neutrons." Phys. Rev. 93 (1954): 268-269.

---. "Time-dependent correlations between spins and neutron scattering in ferromagnetic crystals." Phys. Rev. 95 (1954): 1374-1384.

Warburg, O. "Über Beziehung zwischen Zellstruktur und Biochimischen Reaktionen." Pügers Archiv. 145 (1919): 277-291.

Wheatley, D.N. "Diffusion theory, the cell and the synapse." BioSystems 45 (1998): 151-163.

---. "Diffusion theory in biology." J. Biol. Educ. 27 (1993): 181-188.

---. "Water in life." Nature 366 (1993): 308.

---. "Success and failures on the final frontier." Biologist 44 (1997), 341.

---. "On the vital role of fluid movement in organisms and cells: a brief historical account from Harvey to Coulson, extending the hypothesis of circulation.Med. Hypoth. 52 (1999): 275-284.

---. "Review: Diffusion, perfusion and the exclusion principles in the structural and functional organization of the living cell: reappraisal of the properties of the 'ground substance'." J. Exp. Biol. 206 (2003): 1955-1961.

Wojnar, R. and J. Stecki. "Kinetic equation for the dilute Lorentz gas with attractive forces." Physica A 84 (1976): 316-335.

Wojnar, R. "Nonlinear heat equation and two-level diffusion." Rep. Math. Phys. 49 (2002): 415-426.

Young, J. Zachary. "In doubt and certainty in science." The BBC Reith Lectures 1950. Oxford: Oxford University Press, 1951.

Zwanzig, R. "Statistical mechanics of irreversibility." New York: 1961.

---. "Ensemble method in the theory of irreversibility." J. Chem. Phys. 33 (1960): 1338-1341.

---. "Method for finding the density expansion of transport coefficients of gases." Phys. Rev. 129 (1963): 486-494.

---. "Approximate eigenfunctions of the Liouville operator in classical many-body systems." Phys. Rev. 144 (1966): 170-177.


Full Text: PDF

Download statistics: 2466



e-ISSN: 2300-133X, ISSN: 2081-545X

Since 2017 Open Access in De Gruyter and CrossCheck access cofinanced by The Ministry of Science and Higher Education - Republic of Poland - DUN 775/P-DUN/2017 see more

The Journal is indexed in:
and others see Abstracting and Indexing list

AUPC SM is on the List of the Ministry’s scored journals with 20 points for 2019

Deklaracja dostępności cyfrowej