Summary
The decay of a macroscopic unstable state implies anomalous fluctuations in the amplitudes of the decaying parameters, which are the transient extension of the stationary divergences at the critical point of phase transitions. These decays are best studied, theoretically and experimentally, via the stochastic times of intersection of a given threshold. Besides yielding a closed solvable set of moment equations, the stochastic time approach permits to discriminate the transient fluctuations due to the spread in the initial conditions from those arising from noise along the path. These latter ones limit the validity of the so-called asymptotic approximation. Here we develop a detailed theory including scaling laws and then compare it with experimental measurements in order to show the limit of previous approaches.
Riassunto
Il decadimento di uno stato macroscopico instabile produce fluttuazioni anomale nell’ampiezza del parametro in esame, che sono il corrispettivo in transitorio delle fluttuazioni stazionarie al punto critico di una transizione di fase. Tali decadimenti sono meglio studiati, sia teoricamente che sperimentalmente, scegliendo come parametro significativo il tempo stocastico di attraversamento di una soglia prefissata. Oltre a fornire un insieme chiuso di equazioni esattamente solubili per i momenti, questa scelta permette di discriminare le fluttuazioni dovute ad un’indeterminazione nelle condizioni iniziali da quelle prodotte dal rumore lungo la traiettoria. L’influenza di queste ultime limita la validità della cosiddetta approssimazione asintotica. In questo lavoro si sviluppa una teoria dettagliata, che include leggi di scala, e la si confronta con misure sperimentali allo scopo di mostrare i limiti di precedenti metodi.
Резюме
Распад макроскопического нестабильного состояния подразумевает аномальные флуктуации в амплитудах параметров распада, которые представляют переходный процесс для стационарных флуктуаций в критических точках фазовых переходов. Такие распады хорошо изучены теоретически и экспериментально, с помошью стохастических времен точки пересечения для заданного порога. Помимо получения замкнутой решаемой системы уравнений для моментов, стохастический временной подход позволяет дискриминировать переходные флуктуации, обусловленные неопределенностью начальных условий, от флуктуаций, возникающих из-за шума вдоль траектории. Влияние последних флуктуаций ограничивает применимость так называемого асимптотического приближения. Здесь мы развиваем подробную теорию, включаюшую законы подобия, а затем проводим сравнение с экспериментальными измерениями, чтобы показать ограничения предыдущих подходов.
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References
a) Proceedings of the 1978 Oji Seminar on Nonlinear Nonequilibrium Statistical Mechanics, Kyoto, Prog. Theor. Phys. Suppl.,64 (1978).b) Order and Fluctuations in Equilibrium and Nonequilibrium Statistical Mechanics, edited byG. Nicolis, G. Dewel andJ. W. Turner (New York, N. Y., 1981).c) Dynamical Critical Phenomena, edited byC. P. Enz (New York, N. Y., 1979).d) Pattern Formation by Dynamic Systems and Pattern Recognition, edited byH. Haken (New York, N. Y., 1979).
M. Dubois andP. Bergè:Dynamical Critical Phenomena, edited byC. P. Enz (New York, N. Y., 1979);J. P. Gollub:Pattern Formation by Dynamic Systems and Pattern Recognition, edited byH. Haken (New York, N. Y., 1979).
K. Tomita:Proceedings of the 1978 Oji Seminar on Nonlinear Nonequilibrium Statistical Mechanics, Kyoto, Prog. Theor. Phys. Suppl.,64, 452 (1978).
F. T. Arecchi:Order and Fluctuations in Equilibrium and Nonequilibrium Statistical Mechanics, edited byG. Nicolis, G. Dewel andJ. W. Turner (New York, N. Y., 1981), p. 107.
F. T. Arecchi, V. Degiorgio andB. Querzola:Phys. Rev. Lett.,19, 1168 (1967).
F. T. Arecchi andV. Degiorgio:Phys. Rev. A,3, 1108 (1971).
F. T. Arecchi andA. Politi:Phys. Rev. Lett.,45, 1219 (1980).
H. Risken andH. D. Vollmer:Z. Phys.,204, 270 (1967).
N. G. Van Kampen:J. Stat. Phys.,17, 71 (1977).
R. Kubo:Synergetics, edited byH. Haken (Leipzig, 1973).
M Suzuki:J. Stat. Phys. 16, 477 (1977).
F. Haake:Phys. Rev. Lett.,41, 1685 (1978).
F. De Pasquale andP. Tombesi:Phys. Lett. A,72, 7 (1979).
F. De Pasquale, P. Tartaglia andP. Tombesi:Z. Phys. B,43, 361 (1981).
R. B. Griffiths, C. Y. Weng andJ. S. Langer:Phys. Rev.,149, 301 (1966).
K. Lindenberg andV. Seshadri:J. Chem. Phys.,71, 4075 (1979).
M. Mangel:Physica A (Utrecht),97, 616 (1979).
F. Haake, J. W. Haus andR. Glauber:Phys. Rev. A,23, 3255 (1981).
B. Caroli, C. Caroli andB. Roulet:J. Stat. Phys.,21, 415 (1979).
R. L. Stratonovich:Topics in the Theory of Random Noise, Vol.1 (New York, N. Y., 1963).
We recall that, for the validity of eq. (2.8),Q(z, t) must go to 0 (t→∞) faster thant −m. There are pathological physical situations in which this condition does not hold.
A tentative description of optical tristability was formulated byM. Kitano, J. Yabuzaki andT. Ogawa:Phys. Rev. Lett.,46, 926, (1981), with reference to a three-level system, having degenerate 2-level ground state (1 and 2) and a single excited state (3). States 1 and 2 are coupled, respectively, by right and left polarized radiation to level 3, and selection rules forbid in both cases the complementary coupling. An injection of equal amounts of right and left polarized resonant light induces the system to display mono-, bi- and tristable regions. The main drawback of this model is that it neglects the coherence between levels 1 and 2, always present in physical systems (e.g. alkali atoms). This will be shown in a successive work.
The same result was obtained byB. Caroli, C. Caroli andB. Roulet:Physica A (Utrecht),101, 581 (1980), following a WKB approach. Our derivation results to be much simpler.
Handbook of Mathematical Functions, edited byM. Abramowitz andI. A. Stegun (New York, N. Y., 1972).
A. Schenzle andH. Brand:Phys. Rev. A,20, 1628 (1979).
W. Feller:An Introduction to Probability Theory and its Applications, Vol.2 (New York, N. Y., 1966).
M. Mangel:Phys. Rev. A,24, 3226 (1981).
F. T. Arecchi andA. Politi:Lett. Nuovo Cimento,27, 486 (1980).
M. A. Anisimov, E. E. Gorodetskii andV. M. Zaprudskii:Sov. Phys. Usp.,24, 57 (1981).
Equation (4.13) is the only reasonable normalization for\(T_{ab}^ - \). We do not agree withMangel (17), who does not divide the integral in eq. (4.13) by\(P_{ab}^ - \).
H. A. Kramers:Physica (Utrecht),7, 284 (1940).
R. Gilmore:Phys. Rev. A,20, 2510 (1979).
T. Kawakubo andS. Kabashima:J. Phys. Soc. Jpn.,37, 1199 (1974).
F. T. Arecchi, A. Politi andL. Ulivi:Phys. Lett. A,87, 333 (1982).
F. T. Arecchi andA. Politi:Phys. Lett. A,77, 312 (1980).
J. S. Langer: inFluctuations, Instabilities and Phase Transitions, edited byT. Riste (New York, N. Y., 1975).
K. Binder: inFluctuations, Instabilities and Phase Transitions edited byT. Riste (New York, N. Y., 1975).
P. Grossmann andS. Thomae:J. Stat. Phys., to appear.
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Work partly supported by contract CNR-INO 1981.
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Arecchi, F.T., Politi, A. & Ulivi, L. Stochastic-time description of transitions in unstable and multistable systems. Nuov Cim B 71, 119–154 (1982). https://doi.org/10.1007/BF02721698
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DOI: https://doi.org/10.1007/BF02721698