The time-fractional kinetic equation for the non-equilibrium processes

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.


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 literature [17][18][19] . The time evolution of finding probability P(x, t) can be presented on the one-dimensional space by where R(x, x ′ , t − t ′ ) is the transfer probability kernel from the position x to x ′ . 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 R(x, x ′ , t − t ′ ) by reformulated the path integral formulation based on previous approximations [1][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) ≡ ∞ −∞ dx ′ f (x − x ′ )g(x ′ ) denotes a Fourier convolution of the f and g functions. Dividing by u, after Laplace inversion and differentiation ∂ ∂t we obtain another representation (2) uP(k, u) − W 0 (k) = R(k, u) * P(k, u), Vol In this representation, the kernels W(x, x ′ ) and 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 2 . It is known that, in the non-Markovian process, while the jump length variance 2 is finite, the waiting time T diverges due to the spatial deformations, entropic restrictions or other memory effects.
For a non-Markovian processes �(t) is represented by where Ŵ is Gamma function, τ is the macroscopic relaxation parameter and the exponent γ takes the value between 0 < γ < 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)  Here we briefly summarize that, the non-Markovian dynamics in the continuum limit leads to the timefractional 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 β is the inverse temperature, 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 Langer [8][9][10] , which is an extension of a model proposed by Glauber 21 based on Zwanzig theory 22,23 .
Afterwards, following previous theoretical schema [1][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, ∂ x ) is written as which causes the Kramers-Moyal expansion 24,25 ∂P(x, t) www.nature.com/scientificreports/ where ∂ m ∂x m operates at the same time both W(x → x + δ)P(x, t) and W(x → x + δ) . The sums are over all possible values of the multi-indices m. It should also be noted that we set δ = x ′ − x and x ′ = x + δ 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) 1-5 , and then we can write the probability function P(x, t + �t) as We define the Fourier transform F {P}(k, t + �t) of Eq. (14)  Here, introducing Eqs. (16) into (17) and recognizing that for small 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 ẋ(t ′ ) = (x − x 0 )/�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 −ik(t ′ ), x(t ′ ) 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 )/�t → ∞ . 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 values, the integrations can be obtained as 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.

Scientific Reports
In this simple schema, the number of the channels is given by n = n 1 + n 2 where n 1 denotes channels with energy ε 1 and n 2 admits the open channels with energy ε 2 , respectively. In the statistical framework, the Helmholtz energy of such channel model around equilibrium can be written as 26,27 where x = n 2 /n is the open probability P 0 under external potential V, �(x) corresponds to the number of configurations which is given by ln �(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 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 formalism 26,27 . 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 steadystate. 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 voltagegated ion-channel system that behaves as a non-equilibrium system under cell voltage. We introduce the timefractional 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 kinetics [28][29][30][31][32][33] . www.nature.com/scientificreports/