Abstract
In this study, we consider the non-Markovian dynamics of the generic non-equilibrium kinetic process. We summarize the generalized master equation, the continuous and discrete forms of the time-fractional diffusion equation. Using path integral formulation, we generalized the solutions of the Markovian system to the non-Markovian for the non-equilibrium kinetic processes. Then, we obtain the time-fractional kinetic equation for the non-equilibrium systems in terms of free energy. Finally, we introduce a time-fractional equation to analyse time evolution of the open probability for the deformed voltage-gated ion-channel system as an example.
Similar content being viewed by others
Introduction
A stochastic theory of the kinetics in univariant, non-equilibrium systems that undergoes phase transitions has been formulated in Refs.1,2,3,4,5. In these studies, the path integral formulation of the density distribution and conditional probability was given based on the Markovian master equation6,7 and the kinetic transition probability8,9,10.
Although the path integral approximation for the stochastic Markovian and non-Markovian processes are well known and well-studied topics in physics11,12,13,14,15,16, to the best of our knowledge, the path integral formulation of the non-Markovian dynamics of the non-equilibrium kinetic processes has never been discussed so far. Therefore, in this study, we generalized the path integral solutions of the Markovian systems to the non-Markovian case for the non-equilibrium kinetic system according to the methods presented in Refs.1,2,3,4,5.
Therefore, in this study, we consider a generic non-equilibrium kinetic system with non-Markovian dynamics. Following the formalism in Refs.1,2,3,4,5 we construct the conditional probability for the non-Markovian dynamics by using path integral formulation. Then, we show that the non-Markovian processes in the non-equilibrium system kinetic systems lead to time-fractional kinetic equations. Additionally, we consider the simplest voltage-gated ion-channel system as an example to analyze the time-fractional dynamics of the open probability.
The outline of our paper is given as follow: First, we introduce the non-Markovian master equation. Then, in the following section, we obtain the time-fractional kinetic equation for the non-Markovian dynamics of the non-equilibrium kinetic process. Here, we also introduce the fractional kinetic equation of the deformed voltage-gated ion-channel system. Finally, we review the obtained results and present a brief discussion.
Generalized Master equation
First, we briefly give the information about the master equation for the non-Markovian dynamics which is called as generalized master equation (GME) in the literature17,18,19. The time evolution of finding probability P(x, t) can be presented on the one-dimensional space by
where \(\mathfrak {R}(x,x^{\prime },t-t^{\prime })\) is the transfer probability kernel from the position x to \(x^{\prime }\). As seen from Eq. (1) that GME is very different from Markovian form. Here, we use the probability function of the generalized master equation kernel \(\mathfrak {R}(x,x^{\prime },t-t^{\prime })\) by reformulated the path integral formulation based on previous approximations1,2,3,4.
The Fourier-Laplace transformation of Eq. (1) is written as
where u is the Laplace variable, k is the wave number and \(f(x)*g(x)\equiv \int _{-\infty }^{\infty } dx^{\prime }f(x-x^{\prime })g(x^{\prime })\) denotes a Fourier convolution of the f and g functions. Dividing by u, after Laplace inversion and differentiation \(\frac{\partial }{\partial t}\) we obtain another representation
of the Eq. (1). The new kernel of the master equation is given by \(\tilde{\mathfrak {R}}(x,x^{\prime };u)=\mathfrak {R}(x,x^{\prime };u)/u\), i.e. \(\tilde{\mathfrak {R}}(x,x^{\prime };t)=\int _{0}^{t} dt \mathfrak {R}(x,x^{\prime };t)\) where \(\tilde{\mathfrak {R}}\) can be presented with \(\tilde{\mathfrak {R}}(x,x^{\prime };t) = W(x,x^{\prime }) \Pi (t)\). In this representation, the kernels \(W(x,x^{\prime })\) and \(\Pi\) are responsible for spatial correlations and memory in any stochastic process. These kernels are classified by the finite characteristic waiting time T and the finite jump length variance \(\Sigma ^{2}\). It is known that, in the non-Markovian process, while the jump length variance \(\Sigma ^{2}\) is finite, the waiting time T diverges due to the spatial deformations, entropic restrictions or other memory effects.
For a non-Markovian processes \(\Pi (t)\) is represented by
where \(\Gamma\) is Gamma function, \(\tau\) is the macroscopic relaxation parameter and the exponent \(\gamma\) takes the value between \(0<\gamma <1\). We note that Eq. (4) corresponds to the long-tailed waiting time probability distribution.
In the new situation, the new kernel is given by
The solution of the GME shows a strong dependence on its stochastic history. Therefore, the resulting equation is
Equation (6) includes the defining expression20
where \(_{0}D^{1-\gamma }_{t}\) is the Riemann–Liouville fractional derivative20. Time-fractional master equation can be expressed in te form
Here we briefly summarize that, the non-Markovian dynamics in the continuum limit leads to the time-fractional differential equation Eq. (8). The discrete form of Eq. (8) is given by
Using Eq. (9) we will construct the path integral formulation of the conditional probability function P(x, t).
Time-fractional kinetic equation
It should be noted that the kernel of the spatial correlation for the kinetic processes can be defined in terms of Helmholtz free energy
where \(\beta\) is the inverse temperature, \(\Delta\) is a constant and F is the free energy of the thermodynamic system. This definition of the kinetic transition probability in Eq. (10) suggested by Langer8,9,10, which is an extension of a model proposed by Glauber21 based on Zwanzig theory22,23.
Afterwards, following previous theoretical schema1,2,3,4,5 we construct the path integral definition of the probability function for the non-Markovian dynamics of the generic non-equilibrium kinetic. Thus, we write the master equation in Eq. (9) can be given in the form
where \(H(x,\partial _{x})\) is written as
which causes the Kramers-Moyal expansion24,25
where \(\frac{\partial ^{m}}{\partial x^{m}}\) operates at the same time both \(W (x\rightarrow x+\delta ) P(x,t)\) and \(W (x\rightarrow x+\delta )\). The sums are over all possible values of the multi-indices m. It should also be noted that we set \(\delta = x^{\prime }-x\) and \(x^{\prime }=x+\delta\) in Eqs. (12) and (13) for the convenience.
At this point, by using definition of the derivatives we can arrange the left side of Eq. (11)1,2,3,4,5, and then we can write the probability function \(P(x,t+\Delta t)\) as
We define the Fourier transform \(\mathcal {F} \{P\}(k,t+\Delta t)\) of Eq. (14) as
This of course is represented by
where \(H(x_{0},-ik)\) is obtained from Eq. (13) by replacing \(\partial /\partial x\) with \(-ik\) and x with \(x_{0}\)2,3,4. On the other hand, the inverse Fourier transform of Eq. (16) is given by
Here, introducing Eqs. (16) into (17) and recognizing that for small \(\Delta t\) the curly bracket in Eq. (16) is an exponential, in this case, we obtain
The kernel of Eq. (18) can be defined as
where \(\dot{x}(t^{\prime }) = (x-x_{0})/ \Delta t\). This kernel represents the path integral formulation of the conditional probability for non-Markovian kinetics. We clearly see that the integral argument in Eq. (19) corresponds to Lagrangian of the system, which is given as
where \(H \left( -ik(t^{\prime }), x(t^{\prime }) \right)\) can be read as Hamiltonian. The path integral in Eq. (19) can be defined as the limit of the multiple integral
when \(L=(t-t_{0}) / \Delta t \rightarrow \infty\). We consider here that the transition between the small paths along the trajectory are independent of each other. Now, by using Eq. (10) we can write \(H(-ik,x)\) as
or very small \(\Delta\) values, the integrations can be obtained as
Introducing Eqs. (23) into (21) and integrating over k, we get the time-fractional kernel
The path of extreme probability are those for which \(\int _{0}^{\tau }dt^{\prime } L_{\gamma }(x,\dot{x},k)\) is extremized, namely, time-fractional action integral satisfy the condition
where \(L_{\gamma }(k, x,\dot{x})\) is the time-fractional Lagrangian of the system. Euler-Lagrange equation in the action Eq. (25) can be solved due to k as
The solution of Eq. (26) gives the time-fractional kinetic equation
where \(0<\gamma <1\). As can be seen from Eq. (27) that the time evolution of the stochastic variable x is governed by integro-differential operator. The operator \(_{0}D^{\gamma }_{t}\) points out that the solution of the kinetic equation Eq. (27) possesses Mittag–Leffler function.
An example: the deformed ion-channel systems
In the simplest voltage-gated ion channel system, it is assumed that the channels are located on a two-dimensional membrane. At the equilibrium, the channels are open or closed depending on temperature and membrane voltage. Besides, the status of the channels depends on their intrinsic properties, the state of the channel is mainly determined by the membrane potential at a constant temperature. In these models, for simplicity, it is assumed that the channels are identical and distributed randomly on the membrane surface.
In this simple schema, the number of the channels is given by \(n=n_{1}+n_{2}\) where \(n_{1}\) denotes channels with energy \(\varepsilon _{1}\) and \(n_{2}\) admits the open channels with energy \(\varepsilon _{2}\), respectively. In the statistical framework, the Helmholtz energy of such channel model around equilibrium can be written as26,27
where \(x=n_{2}/n\) is the open probability \(P_{0}\) under external potential V, \(\Omega (x)\) corresponds to the number of configurations which is given by \(\ln \Omega (x)= -n [(1-x) \ln (1-x) + x\ln x]\) and z is the number of charges, and \(e_{0}\) is the charge of electron. It is shown that the open probability can be found from first derivative of the Helmholtz free energy in Eq. (28) as to the variable x. Thus it can be written as
where \(V_{0} = - (\varepsilon _{1} - \varepsilon _{2}) / z e_{0}\) is the critical threshold voltage value. In the case of Markovian dynamics, it is assumed that the half of the channels on the membrane surface are open.
The open probability of the channel system in Eq. (29) is a well-known and well-studied topic in the literature. Furthermore, the kinetic behavior of the voltage gated ion channels was also examined within the framework of the Markovian formalism26,27. However, in the presence of the deformations such as a genetic mutation in the channel, we expect that ion channels have non-Markovian dynamics due to decoupling dynamics as stated in the previous section. As a result, channel conductivity is damaged and the cell may not be able to perform its previous tasks. If the channel kinetics is analyzed within the framework of the non-Markovian formalism, the time-fractional kinetic for the open probability of the ion channel system can be obtained as
As seen from this particular example, one can apply the fractional numerical integration method to Eq. (30) to obtain the time evolution of the open probability for the non-Markovian ion channel system far from steady-state. Hence, the time variation of the open probability and relaxation parameters of the non-Markovian ion channels can be obtained by using Eq. (30). The fractional derivative in Eq. (30) clearly indicates that the solution of the relaxation drastically deviates from the Markovian solution given in Refs.26,27.
Conclusion
In this study, firstly, we briefly introduce the generalized master equation for non-Markovian dynamics. We also present the continuous and discrete forms of the time-fractional diffusion equation in the same section. In the subsequent section, we generalized the path integral solutions of the Markovian system to the non-Markovian for the non-equilibrium kinetic processes. Using path integral formulation we obtain the time-fractional kinetic equation for the non-equilibrium systems in terms of free energy. Here, we consider, as an example, a voltage-gated ion-channel system that behaves as a non-equilibrium system under cell voltage. We introduce the time-fractional kinetic equation for the open probability of the simple ion-channel system.
The non-Markovian dynamics and non-equilibrium behavior of the physical systems are very important two topics in physics. Indeed, many physical systems in nature may have one or both of these properties. Systems can be considered as systems which far from equilibrium that cannot reach equilibrium in very large time scales, and such systems can also be considered as systems that reach equilibrium in short time scales. On the other hand, we know that complex or disordered physical systems generally have non-Markovian dynamics. Therefore, it is very interesting to examine the time evolutions of kinetic systems with both non-equilibrium and Markovian dynamics.
The method presented in the study can be applied to the other physical systems to analyze the time-dependent evolution of the kinetic systems such as the interface dynamics of the phase transitions, kinetic flows, bacterial growth phenomena, other deformed ion channels, internet networks, and chemical kinetics28,29,30,31,32,33.
References
Kubo, R., Matsuo, K. & Kithara, K. Fluctuation and relaxation of macrovariates. J. Stat. Phys. 15, 141 (1973).
Kitahara, K. & Metiu, H. On the path integral representation of stochastic processes. J. Stat. Phys. 15, 141–147 (1976).
Metiu, H., Kitahara, K. & Ross, J. Stochastic theory of the kinetics of phase transitions. J. Chem. Phys. 64, 292 (1976).
Kitahara, K., Metiu, H. & Ross, J. Stochastic theory of cluster growth in homogeneous nucleation. J. Chem. Phys. 63, 3156 (1975).
Graham, R. Path integral formulation of general diffusion processes. Z. Phys. B 26, 281–290 (1977).
Lax, M. Fluctuations from the nonequilibrium steady state. Rev. Mod. Phys. 32, 25–64. https://doi.org/10.1103/RevModPhys.32.25 (1960).
van Kampen, N. G. A power series expansion of the master equation. Can. J. Phys. 39, 551–567. https://doi.org/10.1139/p61-056 (1961).
Langer, J. S. Statistical theory of the decay of metastable states. Ann. Phys. 54, 258–275 (1969).
Langer, J. S. Theory of spinodal decomposition in alloys. Ann. Phys. 65, 53–86 (1971).
Langer, J. S. & Turski, L. A. Hydrodynamic model of the condensation of a vapor near its critical point. Phys. Rev. A 8, 3230–3243. https://doi.org/10.1103/PhysRevA.8.3230 (1973).
Weber, M. F. & Frey, E. Master equations and the theory of stochastic path integrals. Rep. Prog. Phys. 80, 046601. https://doi.org/10.1088/1361-6633/aa5ae2 (2017).
Pesquera, L., Rodriguez, M. & Santos, E. Path integrals for non-Markovian processes. Phys. Lett. A 94, 287–289 (1983).
Hänggi, P. Path integral solutions for non-Markovian processes. Z. Phys. B Condens. Matter 75, 275–281 (1989).
McKane, A. J., Luckock, H. C. & Bray, A. J. Path integrals and non-Markov processes. I. General formalism. Phys. Rev. A 41, 644–656. https://doi.org/10.1103/PhysRevA.41.644 (1990).
Bray, A. J., McKane, A. J. & Newman, T. J. Path integrals and non-Markov processes. II. Escape rates and stationary distributions in the weak-noise limit. Phys. Rev. A 41, 657–667. https://doi.org/10.1103/PhysRevA.41.657 (1990).
Luckock, H. C. & McKane, A. J. Path integrals and non-Markov processes. III. Calculation of the escape-rate prefactor in the weak-noise limit. Phys. Rev. A 42, 1982–1996. https://doi.org/10.1103/PhysRevA.42.1982 (1990).
Metzler, R. & Klafter, J. The random walk’s guide to anomalous diffusion: A fractional dynamics approach. Phys. Rep. 339, 1–77 (2000).
Sokolov, I. M., Klafter, J. & Blumen, A. Fractional kinetics. Phys. Today 55, 48–54 (2002).
Hilfer, R. (ed.) Applications of Fractional Calculus in Physics (World Scientific, 2000).
Oldham, K. B. & Spanier, J. The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order (Academic Press, 1974).
Glauber, R. J. Time-dependent statistics of the ising model. J. Math. Phys. 4, 294 (1963).
Zwanzig, R. Memory effects in irreversible thermodynamics. Phys. Rev. 124, 983–992. https://doi.org/10.1103/PhysRev.124.983 (1961).
Zwanzig, R. On the identity of three generalized master equations. Physica 30, 1109–1123 (1964).
Kramers, H. Brownian motion in a field of force and the diffusion model of chemical reactions. Physica 7, 284–304 (1940).
Moyal, J. E. Stochastic processes and statistical physics. J. R. Stat. Soc. Ser. B 11, 150–210 (1949).
Özer, M. & Erdem, R. Dynamics of voltage-gated ion channels in cell membranes by the path probability method. Physica A 331, 51–60 (2004).
Erdem, R. & Ekiz, C. A kinetic model for voltage-gated ion channels in cell membranes based on the path integral method. Physica A 349, 283–290 (2005).
Cardenas, A. E. & Elber, R. Markovian and non-Markovian modeling of membrane dynamics with milestoning. J. Phys. Chem. B 120, 8208–8216. https://doi.org/10.1021/acs.jpcb.6b01890 (2016).
Goychuk, I. & Hänggi, P. Fractional diffusion modeling of ion channel gating. Phys. Rev. E 70, 051915. https://doi.org/10.1103/PhysRevE.70.051915 (2004).
Barkai, E., Garini, Y. & Metzler, R. Strange kinetics of single molecules in living cells. Phys. Today 65, 29–35. https://doi.org/10.1063/PT.3.1677 (2012).
Suthar, D. L., Habenom, H. & Nisar, K. S. Solutions of fractional kinetic equation and the generalized galué type struve function. J. Interdiscipl. Math. 22, 1167–1184. https://doi.org/10.1080/09720502.2019.1706841 (2019).
Mironov, A. D., Habenom, H., Suthar, D. L. & Gebeyehu, M. Application of laplace transform on fractional kinetic equation pertaining to the generalized galué type struve function. Adv. Math. Phys. https://doi.org/10.1155/2019/5074039 (2019).
Suthar, D. L., Kumar, D. & Habenom, H. Solutions of fractional kinetic equation associated with the generalized multiindex bessel function via laplace transform. Differ. Equ. Dyn. Syst. https://doi.org/10.1007/s12591-019-00504-9 (2019).
Acknowledgements
A part of this work had been completed during visiting Potsdam University. Therefore, E. A. would like to thank Professor Ralf Metzler and Potsdam University for their kind hospitality. E. A. also gratefully thanks the Editorial Board Member and referees for useful comments and constructive suggestions. This study was partially funded by Istanbul University.
Author information
Authors and Affiliations
Contributions
E.A. contributes all to the whole work.
Corresponding author
Ethics declarations
Competing interests
The author declares no competing interests.
Additional information
Publisher's note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Aydiner, E. The time-fractional kinetic equation for the non-equilibrium processes. Sci Rep 11, 20625 (2021). https://doi.org/10.1038/s41598-021-00135-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s41598-021-00135-7
This article is cited by
-
Anomalous kinetic study of atenolol release from ATN@DNA a core-shell like structure
Scientific Reports (2023)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.