Summary
The continuous stochastic formalism for the description of systems with birth and death processes randomly distributed in space is developed with the use of local birth and death operators and local generalization of the corresponding Chapman-Kolmogorov equation. The functional stochastic equation for the evolution of the probability functional is derived and its modifications for evolution of the characteristic functional and the first passage time problem are given. The corresponding evolution equations for equal-time correlators are also derived. The results are generalized then on the exothermic and endothermic chemical reactions. As examples of the particular applications of the results the small fluctuations near stable equilibrium state and fluctuations in monomolecular reactions, Lotka-Volterra model, Schlögl reaction and brusselator are considered. It is shown that the two-dimensional Lotka-Volterra model may exhibit synergetic phase transition analogous to the topological transition of the Kosterlitz-Thouless-Berezinskii type. At the end of the paper some general consequences from stochastic evolution of the birth and death processes are discussed and the arguments on their importance in evolution of populations, cellular dynamics and in applications to various chemical and biological problems are presented.
it]Riassunto
Il formalismo stocastico continuo per la descrizione di sistemi con processi di nascita e di morte distribuiti in modo casuale nello spazio è sviluppato con l’uso degli operatori locali di nascita e di morte e la generalizzazione locale della corrispondente equazione di Chapman-Kolmogorov. Si deriva l’equazione funzionale stocastica per l’evoluzione della funzione di probabilità e si danno le sue modifiche per l’evoluzione della funzione caratteristica ed il problema del primo tempo di passaggio. Si derivano anche le corrispondenti equazioni di evoluzione per correlatori a tempo uguale. Si sono anche generalizzati i risultati sulle reazioni chimiche esotermiche ed endotermiche. Come esempi di particolari applicazioni dei risultati sono considerate le piccole fluttuazioni vicino allo stato di equilibrio stabile e le fluttuazioni nelle reazioni monomolecolari, nel modello di Lokta-Volterra, nella reazione di Schlögl e nel brusselator. Si mostra che il modello bidimensionale di Lokta-Volterra può esibire una transizione di fase sinergica analoga alla transizione topologica del tipo di Kosterlitz-Thouless-Berezinskii. Alla fine del lavoro si discutono alcune conseguenze generali dell’evoluzione stocastica dei processi di nascita e di morte e si presentano gli argomenti sulla loro importanza nell’evohizione delle popolazioni, della dinamica cellulare e nelle applicazioni a vari problemi chimici e biologici.
Резюме
Развит континуальны й стохастический формализм для описан ия эволюции процессов р ождения-гибели со слу чайным распределением в про странстве. Формализм основан на использовали локаль ных операторов рождения-гибели и локальном обобщении соответствующего ур авнения Чепмена-Колмогорова. Выведено функциональное стох астическое уравнени е для эволюции функционал а распределения вероятностей и даны е го модификации для эв олюции характеристическог о функционала и задачи о среднем времени пер вого достижения границ. Вы ведены также соответствующ ие уравнения для эвол юции одновременных корре ляторов. Затем результаты обо бщаются на случай экзотермических и эн дотермических химич еских реакций. В качестве ко нкретных приложений рассмотрены задачи о малых флуктуациях вблизи у стойчивого положени я равновесия, о флуктуациях в моном олекулярных реакциях, модели Лотк и-Вольтерры, реакции Ш лёгля и брюсселяторе. Показано, что в двумер ной модели Лотки-Воль терры может существовать синерг е-тический фазовый переход, анал огичный топологичес кому переходу типа Костер лица-Таулеса-Березин ского. В заключение обсужда ются некоторые общие закономерности стохастической эвол юции процессов рожде ния-гибели и приводятся аргумент ы, показывающие их важн ость для эволюции поп уляций и клеточной динамики, а также в различных хи мических и биологиче ских проблемах.
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References
G. Nicolis andI. Prigogine:Self-Organization in Noneqvdlibrium Systems (Wiley, New York, N. Y., 1977).
H. Haken:Synergetics. An Introduction, 2nd Edition,Springer Series in Synergetics, Vol. 1 (Springer, Berlin, 1978).
D. Mc Quakrie:Supplement Review Series in Applied Probability (Methuen, London, 1976).
H. Haken:Z. Phys. B,20, 413 (1975);C. W. Gabdiner, K.J. Macneil, D. F. Walls andI. S. Matheson:J. Stat. Phys.,14, 307 (1976);C. Van Den Broeck, J. Hauard andM. Malek-Mansour:Physica (Utrecht) A,101, 167 (1980);G. Nicolis:Stochastic Nonlinear Systems in Physics, Chemistry, and Biology inProceedings of the Workshop, Bielefeld, F.R.G., October 5–11, 1980, edited byL. Arnold andE. Lefever (Springer, Berlin, 1981), p. 44.
M. Doi:J. Phys. A,9, 1465, 1479 (1976);Ya. B. Zel’dovich andA.A. Ovchinnikov: Z.Éksp Teor. Fiz.,74, 1588 (1978);A. S. Mikhailov andV. V. Yashin:J. Stat. Phys.,38, 347 (1985);L. Peliti:J. Phys. (Paris),46, 1469 (1985).
G. Parisi andY. Wu:Sci. Sin.,24, 483 (1981).
L. D. Landau andE. M. Lifschits:Quawtwm Mechanics. Nowrelativistic Theory (Pergamon Press, New York, N. Y., 1976).
C. Gardiner:J. Stat. Phys.,15, 451 (1976), andS. Grossman:J. Chem. Phys.,65, 2007 (1976), where the fluctuation corrections of the lowest order were obtained.
M. Lax:Fluctuation and coherence phenomena m classical and quantum physics, inStatistical Physics, Phase Transitions, and Superconductivity, edited byM. Chretien, E. P. Gross andS. Deser (Gordon and Breach, New York, N. Y., 1968);
R. Kubo, K. Matsuo andK. Kitahara:J. Stat. Phys.,9, 51 (1973).
V.I. Kljatskin:Statistical Description of Dynamical Systems with Fluctuating Parameters (Nauka, Moscow, 1975).
N. N. Bogoljubov andD. V. Shirkov:Introduction to the Theory of Quantmn Fields (Nauka, Moscow, 1976);A.N. Vasil’ev:Functional Methods in Quantum Theory and Statistics (LGU, Leningrad, 1976);C. Itztkson andJ. B. Zuber:Quantum Fields Theory (McGraw-Hill, New York, N. Y., 1980);R. P. Feynman andA. R. Hibbs:Quantum Mechanics and Path Integrals (McGraw-Hill, New York, N. Y., 1965).
R.L. Stratonovich:Topics on the Theory of Random Noise, Vol. 1 (Gordon and Breach, New York, N. Y., 1963);E. B. Dynkin:Markov Processes (Springer, New York, N. Y., 1965); very clear derivation may be found also inM. A. Leontovich:Introduction to Thermodynamics. Statistical Physics (Nauka, Moscow, 1983).
N. Rivier andD. M. Duffy:J. Phys. (Paris),43, 293 (1982) ;Numerical Methods in the Study of Critical Phenomena, inProceedings of the Collogues, Carry-le-Rouet, France,June 2–4, 1980, edited byJ. Della Dora, J. Demengeot andB. Lacolle (Springer, Berlin, 1981), p. 132.
J. D. Breit, S. Gupta andA. Zaks:Nucl. Phys. B,233, 61 (1984);J. Alfaro, R. Jengo andN. Parga:Phys. Rev. Lett.,54, 369 (1985); similar ideas were developed previously also inH. Yahata:Prog. Theor. Phys.,51, 2003 (1974).
F. Baras, G. Nicolis, M. Malek-Mansotur andJ.W. Turner:J. Stat. Phys.,32, 1 (1973);F. de Pasquale andA. Mecozzi:Phys. Rev. A,31, 2454 (1985).
V. R. Chechetkin, E. B. Levchenko andA. S. Sigov:J. Phys. D,18, 461 (1985);V. R. Chechetkin andA. S. Sigov:J. Phys. D,18, (in press);
V. R. Chechetkin andA. S. Sigov.Philos. Mag. A (in press).
C. Van Den Broeck:J. Stat. Phys.,31, 467 (1983);C. Van Den Broeck andP. Hänggi:Phys. Rev. A,30, 2609 (1984).
N. G. Van Kampen:J. Stat. Phys.,24, 175 (1981). More general stochastic rules are discussed inV. E. Shapieo andV. M. Loginov:Dynamical Systems Subject to Random Disturbances (Nauka, Novosibirsk, 1983).
P. Hänggi:Z. Phys. B,36, 271 (1980).
E. M. Lifschits andL. P. Pitaevskii:Statistical Physics, Part 2 (Nauka, Moscow, 1973).
L. D. Landau andE. M. Lifshits:Statistical Physics, Part 1 (Pergamon Press, New York, N. Y., 1980).
F. V. Bunkin, N. A. Kirichenko andB. S. Lük’janchuk:Usp. Fiz. Nauh,138, 45 (1982).
A. P. Levanjuk:Z. Éksp. Teor. Fiz.,36, 810 (1959);V. L. Ginzburg:Fiz. Tverd. Tela (Leningrad),2, 2034 (1960).
H. E. Stanley:Introduction to Phase Transitions and Critical Phenomena, 2nd Edition (Oxford Press, London, 1980) ;b) A. Z. Patashinkii andV. L. Pokrovskii :Fluctuation Theory of Phase Transitions, 2nd Edition (Nauka, Moscow, 1982).
M. Lax:Rev. Mod. Phys.,32, 25 (1960).
M. A. Lavrent’ev andB. V. Shabat:Methods of the Theory of Functions from One Complex Variable (Nauka, Moscow, 1973).
E. B. Gledzer, P. V. Delszanskii andA.M. Obukhov:The Systems of Hydrodynamic Type and Their Applications (Nauka, Moscow, 1981).
J. D. Murray:Lectures on Nonlinear-Differential-Equation Models in Biology (Clarendon Press, Oxford, 1977).
G. Nikolis, M. Malek-Mansour, A. Van Nypelseer andK. Kitahara:J. Stat. Phys.,14, 417 (1976);
M. Malek-Mansour andG. Nikolis:J. Stat. Phys.,13, 197 (1976).
V. L. Berezinskii:Z. Éksp. Tear. Fig.,59, 907 (1970);61, 1144 (1971);
J. M. Kosteblitz andD. Thouless:J. Phys. C,6, 1181 (1973) ;J. M. Kosterlitz :J. Phys. C,7, 1046 (1974).
A. T. Winfree:Sciences,175, 634 (1972); 181, 137 (1973);A.M. Zhabotinskii:Concentration Autoseintillations (Nauka, Moscow, 1974);L. S. Polak andA. S. Mikhailov:Self-Organization in Nonequilibrium Physico-Chemical Systems (Nauka, Moscow, 1983).
P. L. Chow andW. C. Tam:Bull. Math. Biol.,38, 643 (1976).
K. J. McNeil andD. F. Wails:J. Stat. Phys.,10, 439 (1974);A. Nitzan, P. Ortoleva, J. Deutch andJ. Ross:J. Chem. Phys.,61, 1056 (1974);S. Grossmann:Stochastic Nonlinear Systems in Physics, Chemistry, and Biology inProceedings of the Workshop, Bielefeld, F.R.G., October 5–11, 1980, edited byL. Arnold andR. Lefever (Springer, Berlin, 1981), p. 213.
F. Shlögl:Z. Phys.,253, 147 (1972).
A. S. Sigov andV. R. Chechetkin:Dokl. Akad. Nauk SSSR,285, 360 (1985).
H. A. Kastrup:Phys. Rep.,101, 1 (1983).
J. Langer:Ann. Phys. (N. Y.),54, 258 (1969).
V. P. Scripov andV. P. Koverda:Spontaneous Crystallization of Supercooled Liquids (Nauka, Moscow, 1984).
A. Nitzan:Phys. Rev. A,17, 1513 (1978).
I. Prigogine andK. Lefever:J. Chem. Phys.,48, 1695 (1968).
R. Schranner, S. Grossmann andP. H. Richter:Z. Phys. B,35, 363 (1979);
Y. Kuramoto andT. Tsuzuki:Prog. Theor. Phys.,54, 60 (1975);
N. G. Van Kampen:Adv. Chem. Phys.,34, 245 (1976);
D. Walgraef, G. Dewel andP. Borckmans:Phys. Rev. A,21, 397 (1980).
A.I. Oparin:The Chemical Origin of Life (Springfield, III., 1964);J. D. Bernal:The Origin of Life (Weidenfeld and Nicolson, London, 1967);C. E. Folsome:The Origin of Life (Freeman, San Francisco, Cal., 1979).
V. I. Vernadskii:Biosphere (Mysl’, Moscow, 1967);Living Matter (Nauka, Moscow, 1978);Selected Topics on Geochemistry (Nauka, Moscow, 1983).
I. I. Shmal’gauzen:Ways and Laws of Evolution Process (Nauka, Moscow, 1983);
Yu. M. Sverezev andV. P. Pasekov:Foundations of Mathematical Genetics (Nauka, Moscow, 1982).
A. J. Lichtenberg andM. A. Lieberman:Regular and Stochastic Motion (Springer, New York, N. Y., 1983).
C. W. F. McClare:J. Theor. Biol.,35, 569 (1972);
D. Markowitz andR. M. Nisbet:J. Theor. Biol.,39, 653 (1973);
S. P. Gabuda andA. F. Rzavin:NMR in Crystallohydrates and Hydrated Proteins (Nauka, Novosibirsk, 1978);
S. P. Gabuda:Confined Water: Facts and Hypothesis (Nauka, Novosibirsk, 1982).
P. Cappuccinelli:Motility of Living Cells (Chapman and Hall, London, 1980).
S. W. Kuffler andJ. G. Nicholls:From Neuron to Brain: a Celluar Approach to the Function of the Nervous System (Sinauer Associates, Inc. Publishers, New York, N. Y., 1976); B. Hill:Annu. Rev. Physiol.,38, 139 (1978); C. F. Stevens: Sci.Am.,241, 54 (1979).
N. Wiener:Cybernetics and Society (Eyre and Spottiswoode, London, 1954), Chapt. 5.
W. Feller:An Introduction to Probability Theory and Its Applications, Vol.1 (Wiley, New York, N. Y., 1970).
Yu. A. Klimontovich:Kinetic Theory of Non-Ideal Plasma and Non-Ideal Qas (Nauka, Moscow, 1975);Statistical Physics (Nauka, Moscow, 1982).
F. Spitzer:Principles of Random Wall (Springer, New York, N. Y., 1976).
V. S. Vladimirov:Equations of Mathematical Physics (Nauka, Moscow, 1976).
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Chechetkin, V.E., Lutovinov, V.S. Continuous stochastic approach to birth and death processes and co-operative behaviour of systems far from equilibrium. Nuov Cim B 95, 1–54 (1986). https://doi.org/10.1007/BF02749000
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DOI: https://doi.org/10.1007/BF02749000