- About this Journal
- Abstracting and Indexing
- Aims and Scope
- Article Processing Charges
- Articles in Press
- Author Guidelines
- Bibliographic Information
- Citations to this Journal
- Contact Information
- Editorial Board
- Editorial Workflow
- Free eTOC Alerts
- Publication Ethics
- Reviewers Acknowledgment
- Submit a Manuscript
- Subscription Information
- Table of Contents
International Journal of Stochastic Analysis
Volume 2011 (2011), Article ID 721352, 25 pages
Study of Thermodynamically Inspired Quantities for Both Thermal and External Colored Non-Gaussian Noises Driven Dynamical System
Department of Chemistry, Visva-Bharati, Santiniketan 731 235, India
Received 23 December 2010; Revised 8 March 2011; Accepted 29 March 2011
Academic Editor: Manuel O. Cáceres
Copyright © 2011 Monoj Kumar Sen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We have studied dynamics of both internal and external noises-driven dynamical system in terms of information entropy at both nonstationary and stationary states. Here a unified description of entropy flux and entropy production is considered. Based on the Fokker-Planck description of stochastic processes and the entropy balance equation we have calculated time dependence of the information entropy production and entropy flux in presence and absence of nonequilibrium constraint (NEC). In the presence of NEC we have observed extremum behavior in the variation of entropy production as function of damping strength, noise correlation, and non-Gaussian parameter (which determine the deviation of external noise behavior from Gaussian characteristic), respectively. Thus the properties of noise process are important for entropy production.
In recent years the stochastic dynamics [1–5] community is becoming increasingly interested to study the role of noise in dissipative dynamical systems, because of its potential applications on various noise-induced phenomena, such as noise-induced phase transition , noise-sustained structures in convective instability , stochastic spatiotemporal intermittency , noise-modified bifurcation , noise-induced traveling waves , noise-induced ordering transition , noise-induced front propagation , stochastic resonance [13–15], coherence resonance [16–19], synchronization [20, 21], clustering , noise-induced pattern formation [23, 24]. In the traditional classical thermodynamics, the specific nature of the stochastic process is irrelevant but it may play an important role on the way to equilibration of a given nonequilibrium state of the noise-driven dynamical system. The relaxation behavior of the stochastic processes can be understood using information entropy (). Now the information entropy becomes a focal theme in the field of stochastic processes [25–28]. In  the authors have been studied the transition from the slow-wave sleep to the rapid-eye-movement sleep in terms of the information entropy. Crochik and Tomé  calculated the entropy production in the majority-vote model and showed that the entropy production exhibits a singularity at the critical point. The time evolution of mainly considers the signature of the rate of phase space expansion and contraction in the random force-driven Brownian motion. This implies that the specific nature of the random process has a strong role to play with . In view of the importance of the characteristics of the frictional and the random forces, the specific nature of the random process has a strong role to play with information entropy flux and entropy production in presence and absence of nonequilibrium constraint. The random force may be of both internal and external origins. We assume that the internal thermal noise is Gaussian in characteristic. But the external noise may be of non-Gaussian properties. Again we will discuss this aspect in the later part. However, the frictional force of thermal environment may be proportional to momentum of triggered particle, or it may associated with finite memory kernel. Stochastic processes with frictional memory kernel (i.e., non-Markovian stochastic processes) are important in many situations such as chemical reactions, isomerization [5, 29–31], and Josephson junction . Extension of Kramers' rate theory to non-Markovian stochastic processes has been the subject of the recent past . We now want to discuss the efficacy of choosing non-Gaussian noise instead of Gaussian one for external environmental random perturbations. Experimental data indicates that the noise in biological processes may have a non-Gaussian character. Examples include, among others, flow of current through voltage-sensitive ion channels in a cell membrane or experiments on the sensory systems of rat skin [33, 34]. Recent detailed studies on the source of fluctuations in different biology systems [35, 36] have clearly established that, in such a context, noise sources are, in general, non-Gaussian. Recently, Fuentes et al.  have shown that the stochastic resonance can be enhanced when the subsystem departs from Gaussian behavior and the system shows marked “robustness” against noise tuning, that is, the signal-to-noise ratio curve can flatten when departing from Gaussian behavior, implying that the system does not require fine tuning of the noise intensity in order to maximize its response to a weak external signal. This theoretical finding was verified experimentally by Castro et al. . Very recently the role of colored non-Gaussian noise having continuous distribution has been investigated in the context of synchronization of coupled phase oscillators [20, 21], kinetics of self-induced aggregation of Brownian particles , escape through an unstable limit cycle [39, 40], escape from a metastable state [41–47], coherent resonance in the noise-driven neurons , and ratchet problem . Furthermore, non-Gaussian noise of third order has been shown to be useful in some of the autocatalytic reactions . The objective of the present paper is the study of time dependence of information entropy production and entropy flux in a unified description for internal and external noises-driven system in the presence and the absence of nonequilibrium constraint.
2. Calculation of the Information Entropy Flux and Production
2.1. Relaxation of the Noise-Driven Dynamical System to a Stationery State
We consider a stochastic process in the presence of both internal thermal noise and external noise. The Langevin equation of motion for this process can be written as Now making use of (2.51) in (2.41), (2.42), and (2.47) we finally obtain explicit time dependence of the entropy flux () and the entropy production () having all order contribution with respect to deviation from equilibrium and the entropy production () due to irreversibility in the process as Thus the above equations describe the time dependence of information entropy production and flux due to irreversibility in the process in the presence of nonequilibrium constraint in a unified scheme for both internal and external noises-driven systems. We now explore explicit dependence of the above thermodynamically inspired quantities on time and properties of the noise. First, in Figure 3 the variation of with time is plotted. It shows that the entropy production first decreases with time and then passes through the minimum and finally reaches the following steady state value: for external Gaussian noise (solid curve). But dashed and dotted curves in this figure imply that the minimum is going to disappear and the new steady state is becoming very close to the original one which is driven by the non equilibrium constraints as the noise behavior deviates more from the Gaussian characteristics.
These observations can be explained by simplifying (2.68) in the limit and as First term in the numerator in (2.71), which vanishes as , implies that the external force increases entropy production while the second term corresponds to the decrease of entropy production with time due to dissipative action. Because of these two opposite effects a system thrown away from a steady state by a small external force relaxes to a new steady state passing through a minimum in entropy production with time.
We now consider long time behavior of (2.68), and (2.69). At (2.68), and (2.69) reduce to the following equations: Equation (2.72) describes why is vanishingly small for external non-Gaussian noise (NGN) at long time for the given parameter set in Figure 3. In the effective non equilibrium constraint term () is smaller for NGN compared to Gaussian noise because is greater for the former than the latter. However, the above (2.72) implies that the system with higher effective diffusion constant is more robust against the given non equilibrium constraint and the entropy production decreases monotonically with increase of temperature of the thermal bath. It is demonstrated in Figure 4. It shows that the rate of decrease is higher for external Gaussian noise compared to non-Gaussian. The change of temperature is less effective in the case of the latter since the effective noise strength for NGN is higher than GN. Now we check whether the above results reduce to the standard result or not. In the absence of external noise (, , and ) (2.72) becomes It implies that the external force is not effective to drive the equilibrium state to a new steady state if . However, in the Markovian limit () (2.74) reduces to which is the standard result for entropy production of irreversible processes for a Brownian particle in thermodynamically closed system.
Using (2.72) and (2.73) one can have another important check also on the above results through We now demonstrate the variation of as a function of damping strength in Figure 5 in the presence of internal non Markovian thermal bath and external noise. Solid and dotted curves are corresponding to external colored Gaussian and non Gaussian noises, respectively. This convention has been followed for the rest of the figures. Both the curves in Figure 5 show extremum behavior as a result of interplay of effective damping , diffusion constant (), and noise correlation time. The entropy production becomes close to zero at the minimum since as well as effective non equilibrium constraint is vanishingly small. Thus for external colored non Gaussian becomes very small at larger damping strength compared to Gaussian noise.
In Figure 6 we have presented how the entropy production depends on noise correlation time of internal colored noise in the presence of external colored noise. There is both maximum and minimum for external Gaussian noise. But the minimum disappears for non Gaussian noise. At certain critical value of , the product may be equal to unity, and then the effective non equilibrium constraint () becomes very small and the external force is not able to drive the stationary state to a new steady state and the minimum appears. The maximum appears when .
In the next step, we have demonstrated the variation of the entropy production as a function of (which is related to correlation time of external noise) in Figure 7. It shows that first decreases with and then passes through the minimum and finally reaches the following limiting value for external Gaussian noise: since at large , and can be approximated as and . However, the minimum appears as a result of similar kind of interplay as mentioned for Figure 6. But the minimum disappears for external non Gaussian noise and monotonically increases to the above limiting value.
Finally, in Figure 8 we have plotted the entropy production versus (which accounts the deviation of noise behavior from Gaussian characteristic). It shows that at some critical value of passes through a minimum as result of interplay of the effective damping and since depends on . Thus the effectiveness of the nonequilibrium constraint depends on the deviation of noise properties from Gaussian characteristics.
In conclusion, we have considered the relaxation behavior of a given nonequilibrium state of the thermal broad band noise-driven harmonic oscillator in presence and absence of nonequilibrium constraint. Here we have studied the time dependence of information entropy production and entropy flux based on the Fokker-Planck description of noise process and the entropy balance equation. It includes the following points.(1) Entropy production monotonically decreases with time to a stationary value in the absence of the non equilibrium constraint (NEC). But in the presence of NEC it first decreases with time and then increases passing through a minimum and finally reaches a limiting value for external Gaussian noise for a given parameter set. But the minimum is going to disappear as the noise behavior deviates more from the Gaussian characteristics.(2) It is difficult for the non equilibrium constraint to drive the equilibrium state to a steady state as the temperature of the thermal bath increases and the rate of decreases of the entropy production with temperature is fast for external colored Gaussian noise compared to non Gaussian one.(3) In the presence of NEC we have observed extremum behavior in the variation of entropy production as function of damping strength, noise correlation, non Gaussian parameter (which determine the deviation of external noise behavior from Gaussian characteristic), respectively. Thus the properties of noise process are important for entropy production.
To be mentioned here is that our present calculations are, of course, restricted to the harmonic oscillator (HO). However, insights of this important system usually have a wide impact, as the HO constitutes much more than a mere example. In general Kramers' problem on barrier crossing dynamics is studied analytically by linearization of the nonlinear potential energy function around the fixed points . Qualitatively one can say that greater entropy production of a system implies that the barrier crossing rate is larger since the former increases with increase of phase space expansion rate. Thus we hope that our present study will be useful for the understanding of the various phenomena in colored noise-driven thermodynamically closed systems. Another point to be mentioned is that one can generalize the present study considering more complex cases, such as a thermal environment having non exponential decaying memory kernel. Also one can generalize it to have an exact study for external colored nonGaussian noise.
More about the Linear Transformation
Here we have shown that the linear transformation (2.10) used in Section 2 can also be applied directly to the Langevin dynamics described by (2.1), (2.5), (2.6), and (2.7) to derive the Fokker-Planck equation (2.11). Multiplying in both sides of (2.1), (2.5), and (2.6), respectively, and then adding all the equations (2.1), (2.5), (2.6), and (2.7) we have This is the Langevin equation of motion corresponding to the Fokker-Planck equation (2.11). The above equation implies that (2.1), (2.5), (2.6), and (2.7) are the projection of . In the weak noise limit it becomes The solution of this equation is The effective damping constant in the above equation is finite for the finite value of , , and , and it does not correspond to a particular eigenvalue of the matrix formed by the deterministic parts in the right-hand side of (2.1), (2.5), (2.6), and (2.7). Equations (2.1), (2.5), (2.6), and (2.7) can be written in matrix notation as follows: where The above discussion implies that does not correspond to a particular eigenvalue of the matrix . However, we now come back to (A.3). It implies that is finite at a finite time , and it is not the slow variable of the original dynamics since it satisfies the initial condition taking contribution of all the variables of the phase space and is not the smallest eigenvalue of the matrix . Thus in (A.3) considers contribution from all the variables at arbitrary time. Hence the linear transformation (2.10) being used to reduce the Fokker-Planck equation (2.9) into (2.11) works at any time.
Thanks are due to Council of Science and Industrial Research for partial financial support.
- N. Wax, Selected Papers on Noise and Stochastic Process, Dover, New York, NY, USA, 1954.
- R. L. Stratonovich, Topics in the Theory of Random Noise, Gordon and Breach, New York, NY, USA, 1963.
- C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry, and the Natural Science, Springer, Berlin, Germany, 1983.
- H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications, Springer, Berlin, Germany, 1984.
- P. Hänggi, P. Talkner, and M. Borkovec, “Reaction-rate theory: fifty years after Kramers,” Reviews of Modern Physics, vol. 62, no. 2, pp. 251–341, 1990.
- J. Garcia-Ojalvo and J. M. Sancho, Noise in Spatially Extended Systems, Springer, New York, NY, USA, 1999.
- G. Agez, P. Glorieux, M. Taki, and E. Louvergneaux, “Two-dimensional noise-sustained structures in optics: theory and experiments,” Physical Review A, vol. 74, no. 4, Article ID 043814, 2006.
- M. G. Zimmermann, R. Toral, O. Piro, and M. San Miguel, “Stochastic spatiotemporal intermittency and noise-induced transition to an absorbing phase,” Physical Review Letters, vol. 85, no. 17, pp. 3612–3615, 2000.
- G. Agez, M. G. Clerc, and E. Louvergneaux, “Universal shape law of stochastic supercritical bifurcations: theory and experiments,” Physical Review E, vol. 77, no. 2, 2008.
- L. Q. Zhou, X. Jia, and Q. Ouyang, “Experimental and numerical studies of noise-induced coherent patterns in a subexcitable system,” Physical Review Letters, vol. 88, no. 13, Article ID 138301, pp. 1–4, 2002.
- R. Müller, K. Lippert, A. Kühnel, and U. Behn, “First-order nonequilibrium phase transition in a spatially extended system,” Physical Review E, vol. 56, no. 3, pp. 2658–2662, 1997.
- M. G. Clerc, C. Falcon, and E. Tirapegui, “Additive noise induces front propagation,” Physical Review Letters, vol. 94, no. 14, Article ID 148302, 2005.
- R. Benzi, G. Parisi, A. Sutera, and A. Vulpiani, “Stochastic resonance in climatic change,” Tellus, vol. 34, p. 10, 1982.
- L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni, “Stochastic resonance,” Reviews of Modern Physics, vol. 70, p. 223, 1998.
- H. Calisto, F. Mora, and E. Tirapegui, “Stochastic resonance in a linear system: an exact solution,” Physical Review E, vol. 74, no. 2, Article ID 022102, 2006.
- A. S. Pikovsky and J. Kurths, “Coherence resonance in a noise-driven excitable system,” Physical Review Letters, vol. 78, no. 5, pp. 775–778, 1997.
- M. O. Magnasco, “Forced thermal ratchets,” Physical Review Letters, vol. 71, no. 10, pp. 1477–1481, 1993.
- F. Jülicher, A. Ajdari, and J. Prost, “Modeling molecular motors,” Reviews of Modern Physics, vol. 69, no. 4, pp. 1269–1281, 1997.
- P. Reiman, “Brownian motors : noisy transport far from equilibrium,” Physics Reports, vol. 361, p. 57, 2002.
- B. C. Bag, K. G. Petrosyan, and C.-K. Hu, “Influence of noise on the synchronization of the stochastic Kuramoto model,” Physical Review E, vol. 76, no. 5, Article ID 056210, 6 pages, 2007.
- M. K. Sen, B. C. Bag, K. G. Petrosyan, and C.-K. Hu, “Effect of time delay on the onset of synchronization of the stochastic Kuramoto model,” Journal of Statistical Mechanics, vol. 2010, no. 8, 2010.
- P. K. Ghosh, M. K. Sen, and B. C. Bag, “Kinetics of self-induced aggregation of Brownian particles: non-Markovian and non-Gaussian features,” Physical Review E, vol. 78, no. 5, Article ID 051103, 2008.
- J. Buceta, M. Ibañes, J. M. Sancho, and K. Lindenberg, “Noise-driven mechanism for pattern formation,” Physical Review E, vol. 67, no. 2, Article ID 021113, pp. 1–8, 2003.
- S. Dutta, S. S. Riaz, and D. S. Ray, “Noise-induced instability: an approach based on higher-order moments,” Physical Review E, vol. 71, no. 3, Article ID 036216, pp. 1–7, 2005.
- D. Osorio-González, M. Mayorga, J. Orozco, and L. Romero-Salazar, “Entropy and thermalization of particles in liquids,” Journal of Chemical Physics, vol. 118, no. 15, pp. 6989–6996, 2003.
- B. Q. Ai, X. J. Wang, G. T. Liu, and L. G. Liu, “Correlated noise in a logistic growth model,” Physical Review E, vol. 67, Article ID 022903, 2003.
- M. L. Steyn-Ross, D. A. Steyn-Ross, J. W. Sleigh, M. T. Wilson, and L. C. Wilcocks, “Proposed mechanism for learning and memory erasure in a white-noise-driven sleeping cortex,” Physical Review E, vol. 72, no. 6, Article ID 061910, 2005.
- L. Crochik and T. Tomé, “Entropy production in the majority-vote model,” Physical Review E, vol. 72, no. 5, Article ID 057103, pp. 1–4, 2005.
- S. P. Velsko, D. H. Waldeck, and G. R. Fleming, “Breakdown of Kramers theory description of photochemical isomerization and the possible involvement of frequency dependent friction,” The Journal of Chemical Physics, vol. 78, no. 1, pp. 249–258, 1982.
- G. Rothenberger, D. K. Negus, and R. M. Hochstrasser, “Solvent influence on photoisomerization dynamics,” The Journal of Chemical Physics, vol. 79, no. 11, pp. 5360–5367, 1983.
- J. Troe, “Elementary reactions in compressed gases and liquids: from collisional energy transfer to diffusion control,” Journal of Physical Chemistry, vol. 90, no. 3, pp. 357–365, 1986.
- D. Esteve, M. H. Devoret, and J. M. Martinis, “Effect of an arbitrary dissipative circuit on the quantum energy levels and tunneling of a Josephson junction,” Physical Review B, vol. 34, no. 1, pp. 158–163, 1986.
- S. M. Bezrukov and I. Vodyanoy, “Noise-induced enhancement of signal transduction across voltage-dependent ion channels,” Nature, vol. 378, no. 6555, pp. 362–364, 1995.
- D. Nozaki, D. J. Mar, P. Grigg, and J. J. Collins, “Effects of colored noise on stochastic resonance in sensory neurons,” Physical Review Letters, vol. 82, no. 2–11, pp. 2402–2405, 1999.
- A. Manwani and C. Koch, “Detecting and estimating signals in noisy cable structures—I: neuronal noise sources,” Neural Computation, vol. 11, no. 8, pp. 1797–1829, 1999.
- A. Manwani, , Ph.D. thesis, Caltech, 2000.
- M. A. Fuentes, R. Toral, and H. S. Wio, “Enhancement of stochastic resonance: the role of non Gaussian noises,” Physica A, vol. 295, no. 1-2, pp. 114–122, 2001.
- F. J. Castro, M. N. Kuperman, M. Fuentes, and H. S. Wio, “Experimental evidence of stochastic resonance without tuning due to non-Gaussian noises,” Physical Review E, vol. 64, no. 5, Article ID 051105, pp. 1–3, 2001.
- B. C. Bag and C.-K. Hu, “Escape through an unstable limit cycle driven by multiplicative colored non-Gaussian and additive white Gaussian noises,” Physical Review E, vol. 75, no. 4, Article ID 042101, 2007.
- M. K. Sen, A. Baura, and B. C. Bag, “Noise induced escape through an unstable limit cycle in the presence of fluctuating barrier,” Journal of Statistical Mechanics, p. 11004, 2009.
- B. C. Bag, “Colored non-Gaussian noise driven systems: mean first passage time,” European Physical Journal B, vol. 34, no. 1, pp. 115–118, 2003.
- P. Majee, G. Goswami, and B. C. Bag, “Colored non-Gaussian noise induced resonant activation,” Chemical Physics Letters, vol. 416, no. 4–6, pp. 256–260, 2005.
- B. C. Bag and C.-K. Hu, “Escape through an unstable limit cycle: resonant activation,” Physical Review E, vol. 73, no. 6, Article ID 061107, 2006.
- G. Goswami, P. Majee, P. Kumar Ghosh, and B. C. Bag, “Colored multiplicative and additive non-Gaussian noise-driven dynamical system: mean first passage time,” Physica A, vol. 374, no. 2, pp. 549–558, 2007.
- G. Goswami, P. Majee, and B. C. Bag, “Escape through a fluctuating energy barrier in the presence of non-gaussian noise,” Fluctuation and Noise Letters, vol. 7, no. 2, pp. L151–L161, 2007.
- M. K. Sen and B. C. Bag, “Generalization of barrier crossing rate for coloured non Gaussian noise driven open systems,” European Physical Journal B, vol. 68, no. 2, pp. 253–259, 2009.
- A. Baura, M. K. Sen, G. Goswami, and B. C. Bag, “Colored non-Gaussian noise driven open systems: Generalization of Kramers’ theory with a unified approach,” Journal of Chemical Physics, vol. 134, p. 044126, 2011.
- Y. Gong, Y. Xie, and Y. Hao, “Coherence resonance induced by the deviation of non-Gaussian noise in coupled Hodgkin-Huxley neurons,” Journal of Chemical Physics, vol. 130, no. 16, Article ID 165106, 2009.
- B. C. Bag and C.-K. Hu, “Current inversion induced by colored non-Gaussian noise,” Journal of Statistical Mechanics, vol. 2009, no. 2, article P02003, 2009.
- C. W. Gradiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Processes, Springer, New York, NY, USA, 1985.
- S. A. Adelman, “Fokker–Planck equations for simple non-Markovian systems,” Journal of Chemical Physics, vol. 64, p. 124, 1976.
- B. Bagchi and D. W. Oxtoby, “The effect of frequency dependent friction on isomerization dynamics in solution,” The Journal of Chemical Physics, vol. 78, no. 5, pp. 2735–2741, 1983.
- J. P. Hansen and I. R. McDonald, Theory of Simple Liquids, Academic Press, London, UK, 1976.
- S. Okuyama and D. W. Oxtoby, “Non-Markovian dynamics and barrier crossing rates at high viscosity,” The Journal of Chemical Physics, vol. 84, no. 1, pp. 5830–5835, 1985.
- K. Lindenberg and B. J. West, The Nonequilibrium Statistical Mechanics of Open and Closed Systems, VCH, New York, NY, USA, 1990.
- J. K. Bhattacharjee, Statistical Physics: Equilibrium and non Equilibrium Aspects, Allied Publishers, New Delhi, India, 1997.
- D. Holcman and M. Tsodyks, “The emergence of up and down states in cortical networks,” PLoS Computational Biology, vol. 2, no. 3, pp. 174–181, 2006.
- P. Dayan and F. L. Abbott, Theoretical Neuroscience: Computational and Mathmatical modleing of Neural Systems, The MIT Press, Cambridge, Mass, USA, 2005.
- H. A. Kramers, “Brownian motion in a field of force and the diffusion model of chemical reactions,” Physica, vol. 7, no. 4, pp. 284–304, 1940.
- J. S. Langer, “Statistical theory of the decay of metastable states,” Annals of Physics, vol. 54, p. 258, 1969.
- B. C. Bag and D. S. Ray, “Fluctuation-dissipation relationship in chaotic dynamics,” Physical Review E, vol. 62, no. 2 B, pp. 1927–1935, 2000.
- L. Borland, “Ito-Langevin equations within generalized thermostatistics,” Physics Letters, Section A, vol. 245, no. 1-2, pp. 67–72, 1998.
- M. A. Fuentes, H. S. Wio, and R. Toral, “Effective Markovian approximation for non-Gaussian noises: a path integral approach,” Physica A, vol. 303, no. 1-2, pp. 91–104, 2002.
- C. R. Doering and J. C. Gadoua, “Resonant activation over a fluctuating barrier,” Physical Review Letters, vol. 69, no. 16, pp. 2318–2321, 1992.
- E. T. Jaynes, “New Engineering Applications of Information Theory,” in Statistical Physics, W. K. Ford, Ed., Benjamin, New York, NY, USA, 1963.
- A. Katz, Statistical Mechanics, Freemann, San Francisco, Calif, USA, 1967.
- D. Daems and G. Nicolis, “Entropy production and phase space volume contraction,” Physical Review E, vol. 59, no. 4, pp. 4000–4006, 1999.
- G. E. Uhlenbech and L. S. Ornstein, “On the Theory of the Brownian Motion,” Physical Review, vol. 36, p. 823, 1930.
- S. Chandrasekhar, Reviews of Modern Physics, vol. 15, p. 1, 1943.
- W. H. Louisell, Quantum Statistical Properties of Radiation, John Wiley & Sons, New York, NY, USA, 1990.