First-passage time and change of entropy

Abstract

The first-passage time is proposed as an independent thermodynamic parameter of the statistical distribution that generalizes the Gibbs distribution. The theory does not include the determination of the first-passage statistics itself. A random process is set that describes a physical phenomenon. The first-passage statistics is determined from this random process. The thermodynamic parameter conjugated to the first-passage time is the same as the Laplace transform parameter of the first-passage time distribution in the partition function. The corresponding partition function is divided into multipliers, one of which is associated with the equilibrium parameters, and the second one with the parameters of the first-passage time distribution. The thermodynamic parameter conjugated to the first-passage time can be expressed in terms of the deviation of the entropy from the equilibrium value. Thus, all moments of the distribution of the first-passage time are expressed in terms of the deviation of the entropy from its equilibrium value and the external forces acting on the system. By changing the thermodynamic forces, it is possible to change the first-passage time.

Graphic abstract

Appendix A: Feller process: possibility of increasing the average FPT

The described approach is applicable to multiple problems with FPT concept. For example, in [82], using FPTs, problems such as the thermal motion of a small tracer in a viscous medium, adhesion bond dissociation under mechanical stress, algorithmic trading, first crossing of a moving boundary by Brownian motion, and quadratic double-well potential are considered. In [82], the random variable $\tau =\mathrm{{inf}}\{ t> a: X(t)> L\}$ was considered, describing the first exit time of a stochastic process X(t) outside from the interval [-L, L] with starting point$x_{0}$ at $t_{0}=$0. The expression for the Laplace transform of the FPT distribution used in [82] is more complicated than (19) and has the form

$$\begin{aligned} Z_{\gamma } (r_{0} ,\gamma )=\frac{U(\frac{\gamma L^{2}}{4\kappa D},\frac{d}{2},\kappa \frac{r_{0}^{2}}{L^{2}})}{U(\frac{\gamma L^{2}}{4\kappa D},\frac{d}{2},\kappa )}\frac{U(0,\frac{d}{2},\kappa )}{U(0,\frac{d}{2},\kappa \frac{r_{0}^{2}}{L^{2}})} \end{aligned}$$

(36)

(the same expression (36) was obtained in [26] for Feller processes), where

$$\begin{aligned} U(a,b;z)= & {} \frac{\Gamma (1-b)}{\Gamma (a-b+1)}M(a,b;z)\nonumber \\&+\frac{\Gamma (b-1)}{\Gamma (a)}z^{1-b}M(a-b+1,2-b;z) \end{aligned}$$

(37)

is the confluent hypergeometric function of the second kind [100] (also known as Tricomi function), $M(a,b;z)=_{1}F_{1}$(a,b,z)$=\sum \nolimits _{n=0}^\infty {\frac{a^{(n)}z^{n}}{b^{(n)}n!}}$ is Kummer function, the confluent hypergeometric function of the first kind [100] with $a^{(0)}=1\, {,\, \mathrm{and}\, }\,a^{(n)}=a(a+1) \cdots (a+n-1)=\Gamma (a+n)/\Gamma (a)$, where $\Gamma (z)$ is the gamma function. In (36) $\kappa =kL^{2}/2k_\mathrm{{B}} T$. In [82], a diffusing particle of mass m trapped by a harmonic potential of strength k and pulled by a constant force $F_{0}$ is considered; the force in the Langevin equation $F(X(t))=$-kX(t)$+F_{0}$ includes the externally applied Hooke term and constant force $F_{0}$, $D=k_\mathrm{{B}}T/\varsigma$ is the diffusion coefficient, and $\varsigma$ is the drag constant.

An equation of the form (17) for $\gamma (\Delta )$ and Laplace transform (36) takes on a more complex form and it is generally difficult to find an analytical solution. However, in some approximations [82], a closed analytical expression to the problem can be found.

Consider the Feller process, which is a diffusion process with linear drift and linear diffusion coefficient vanishing at zero point of coordinates. In [26], an expression was obtained for the   Laplace  transform  of  the  probability  density  of  the  the  Feller  process  Y(t)   to  reach  the  boundary  ${x\, =\, xc}$   for  ${x>xc}$ for  the  first  time  (x   is  the  initial  value  of  the  process  at  ${t=0}$) in the form

$$\begin{aligned} Z_{\gamma } =\frac{U(\gamma ,b;x)}{U(0 ,b;x_{c} )}\frac{U(\gamma ,b;x_{c} )}{U(0,b;x)}. \end{aligned}$$

(38)

This  expression  (38)  differs  from  (36)  only  in  the  values  of  the  parameters.  In  (38),  expression  (37)  with  parameters  ${a=\gamma }$,  ${b=2\beta _1/k}^{2}$,  where  ${\beta _1}$,  $k>0$  are  drift  and  diffusion  parameters  of  the  Feller  process.  The  time  evolution  of  the  process  is  thus  governed  by 

$$\begin{aligned} dY(t)=[-\alpha Y(t)+\beta _{1} ]\mathrm{{d}}t+k\sqrt{Y(t)} \mathrm{{d}}W(t), \end{aligned}$$

where $\alpha >0$, $\beta _{1}$, k are constant parameters, $W(t)_{\, }$is the Wiener process. In the approximation of small values of $a = \gamma$ in (37), (38), one can use small $\gamma$ expansion of the Kummer function [26]

$$\begin{aligned} U(a,b,x)= & {} 1+aU_{1} (x)+O(a^{2}),\quad \\ U_{1} (x)= & {} -\Psi (1-b)-\int \limits _0^x U(1,1+b,z)\mathrm{{d}}z,\\ \Psi (x)= & {} \frac{{\Gamma }'(x)}{\Gamma (x)},\quad 0<b<1. \end{aligned}$$

In this approximation at $x_{c}=$ 0

$$\begin{aligned} Z_{1\gamma }= & {} \frac{1+\gamma U_{1} (x)}{1+\gamma U_{1} (0)},\quad \nonumber \\ \bar{{T}}_{\gamma }= & {} \frac{U_{1} (0)}{1+\gamma U_{1} (0)}-\frac{U_{1} (x)}{1+\gamma U_{1} (x)},\nonumber \\ T_{0}= & {} U_{1} (0)-U_{1} (x)=\int \limits _0^x {U(1,1+b,z)\mathrm{{d}}z} . \end{aligned}$$

(39)

An increase of the mean FPT, when $\bar{{T}}_{\gamma } >T_{0}$, in this case is possible at

$$\begin{aligned} U_{1} (0)<T_{0}<2U_{1} (0),\quad \quad U_{1} (0)=-\Psi (1-b)>0,\quad 0<b<1. \nonumber \\ \end{aligned}$$

(40)

We assume that $\gamma > 0$. Consider two cases. In the first case $1+\gamma U_{1} (x)=1-\gamma (T_{0} -U_{1} (0))>0$, $T_{0} -U_{1} (0)>0,\quad U_{1} (x)=U_{1} (0)-T_{0} <0$. Then, the values of the parameter $\gamma$ for which the inequality $\bar{{T}}_{\gamma } >T_{0}$ for the quantities from (39) is satisfied lie in the interval

$$\begin{aligned} \frac{[2U_{1} (0)-T_{0} ]}{U_{1} (0)[T_{0} -U_{1} (0)]}<\gamma <\frac{1}{[T_{0} -U_{1} (0)]}. \end{aligned}$$

(41)

For the case $1+\gamma U_{1} (x)=1-\gamma (T_{0} -U_{1} (0))<0,\quad$, the opposite inequality $[2U_{1} (0)-T_{0} ]/U_{1} (0)[T_{0} -U_{1} (0)]>\gamma >1/[T_{0} -U_{1} (0)]$ should hold. However, it is not the case, since from (44), it follows that $0<2-T_{0} /U_{1} (0)<1$.

The fulfillment of inequality (41) depends on the parameters b and x. Equation (17) for determining the dependence of $\gamma$ on $\Delta$ taking into account the term of the form (10), (11), (16) in this case takes the form

$$\begin{aligned}&-\Delta =\ln (1+\gamma U_{1} (x))-\ln (1+\gamma U_{1} (0))\\&\quad +\frac{\gamma [U_{1} (0)+\beta \partial U_{1} (0)/\partial \beta ]}{1+\gamma U_{1} (0)}-\frac{\gamma [U_{1} (x)+\beta \partial U_{1} (x)/\partial \beta ]}{1+\gamma U_{1} (x)} . \end{aligned}$$

Expanding the right-hand side of this equation in a series in $\gamma$, and taking into account the terms up to $\gamma ^{2}$, we obtain the quadratic equation

$$\begin{aligned}&\gamma ^{2}a_{2} +\gamma b_{2} -\Delta =0, \nonumber \\&a_{2}=(U_{1} ^{2}(0)-U_{1}^{2}(x))/2+U_{1} (0)\beta \partial U_{1} (0)/\partial \beta -U_{1} (x)\beta \partial U_{1} (x)/\partial \beta ,\nonumber \\&a_{2} =T_{0} \bigg [\frac{1}{2}(2U_{1} (0)-T_{0} )+\beta \frac{\partial U_{1} (0)}{\partial \beta }\bigg ]+[U_{1} (0)-T_{0} ]\beta \frac{\partial T_{0} }{\partial \beta }, \nonumber \\&b_{2}=-\beta \left( \frac{\partial U_{1} (0)}{\partial \beta }-\frac{\partial U_{1} (x)}{\partial \beta }\right) \nonumber \\&\quad =-\beta \frac{\partial T_{0} }{\partial \beta }, \frac{\partial U_{1} (0)}{\partial \beta }\nonumber \\&\quad =\frac{\partial U_{1} (0)}{\partial b}\frac{\partial b}{\partial \beta },\quad \frac{\partial U_{1} (x)}{\partial \beta }=\frac{\partial U_{1} (x)}{\partial b}\frac{\partial b}{\partial \beta },\quad \beta =\frac{1}{T},\nonumber \\&b_{2} =-\beta \frac{\partial b}{\partial \beta }\int \limits _0^x \frac{\partial U(1,1+b,z)}{\partial b}\mathrm{{d}}z,\quad U(1,1+b,z)\nonumber \\&\quad =\frac{\Gamma (-b)}{\Gamma (1-b)}M(1,1+b,z)+\Gamma (b)z^{-b}e^{z} . \end{aligned}$$

(42)

For the solution of the quadratic equation (42), the condition (41) takes the form

$$\begin{aligned}&\frac{[2U_{1} (0)-T_{0} ]}{U_{1}^{2}(0)[T_{0} -U_{1} (0)]^{2}}\bigg \{T_{0} [2U_{1} (0)-T_{0} ]\bigg [\frac{1}{2}(2U_{1} (0)-T_{0} )\nonumber \\&\quad +\beta {{\partial U_{1} (0)} \over {\partial \beta }}\bigg ]+[U_{1} (0)-T_{0} ][3U_{1} (0)-T_{0} ]\beta {{\partial T_{0} } \over {\partial \beta }}\bigg \}\nonumber \\&\quad<\Delta <\frac{1}{[T_{0} -U_{1} (0)]^{2}}\bigg \{T_{0} \bigg [\frac{1}{2}(2U_{1} (0)-T_{0} )\nonumber \\&\quad +\beta {{\partial U_{1} (0)} \over {\partial \beta }}\bigg ]\nonumber \\&\quad +2[U_{1} (0)-T_{0} ]\beta {{\partial T_{0} } \over {\partial \beta }}\bigg \}. \end{aligned}$$

(43)

In (42)–(43) do not include explicit expressions for $\partial U_{1} (0) /\partial \beta , \partial T_{0} /\partial \beta$. Now, let us find them. Suppose the parameter $\alpha$ in the stochastic equation for the Feller process is $\alpha =1/T_{0}$. The drift parameter is $f=-\alpha Y+\beta _{1}$. As above, in (18)–(19), we set the parameter f to be $f=L/T_{0}$, where $L=x-x_{c}$. For $x_{c}=$0, $L=x$, where x is the initial value of the process at the initial time moment. Then, $\beta _{1} =f+\alpha Y=(x+Y)/T_{0}$. The diffusion coefficient is $D=k^{2}Y$, and the parameter b of (37)–(40) is equal to$_{\, }b=2\beta _{2} /k^{2}=2(x+Y)Y/T_{0} D$. At $\partial D/\partial \beta =-E_{a} D$

$$\begin{aligned} \frac{\partial b}{\partial \beta }=b\left( E_{a} -\frac{1}{T_{0} }\frac{\partial T_{0} }{\partial \beta }\right) . \end{aligned}$$

(44)

From (39) to (40), we obtain

$$\begin{aligned} \frac{\partial T_{0} }{\partial \beta }=\frac{\partial U_{1} (0)}{\partial \beta }-\frac{\partial U_{1} (x)}{\partial \beta },\quad \frac{\partial U_{1} (0)}{\partial \beta }=\frac{\partial \Psi (1-b)}{\partial (1-b)}\frac{\partial b}{\partial \beta }. \end{aligned}$$

From (11), (39), we find $u_{\gamma } =\frac{\gamma \partial U_{1} (0)/\partial \beta }{1+\gamma U_{1} (0)}-\frac{\gamma \partial U_{1} (x)/\partial \beta }{1+\gamma U_{1} (x)}$. As above, we assume $u_{\gamma } =0$. From here

$$\begin{aligned}&\frac{\partial T_{0} }{\partial \beta }=\frac{\partial \Psi \gamma T_{0} bE_{a} }{1+\gamma (U_{1} (0)+b\partial \Psi )},\quad \partial \Psi =\frac{\partial \Psi (1-b)}{\partial (1-b)},\nonumber \\ \end{aligned}$$

(45)

$$\begin{aligned}&\frac{\partial U_{1} (0)}{\partial \beta }=\frac{\partial \Psi (1+\gamma U_{1} (0))bE_{a} }{1+\gamma (U_{1} (0)+b\partial \Psi )},\quad \nonumber \\&\frac{\partial U_{1} (x)}{\partial \beta }=\frac{\partial \Psi (1+\gamma U_{1} (x))bE_{a} }{1+\gamma (U_{1} (0)+b\partial \Psi )}. \end{aligned}$$

(46)

Substituting (45), (46) into (42), we obtain

$$\begin{aligned} a_{2}= & {} \frac{1}{2}(U^{2}_{1} (0)-U^{2}_{1} (x))+\frac{b\beta E_{a} \partial \Psi }{1+\gamma (U_{1} (0)+b\partial \Psi )}\\&\times [T_{0} +\gamma (U^{2}_{1} (0)-U^{2}_{1} (x))],\\&b_{2} =-\beta \frac{\partial T_{0} }{\partial \beta } =\frac{\gamma \partial \Psi T_{0} b\beta E_{a} }{1+\gamma (U_{1} (0)+b\partial \Psi )}. \end{aligned}$$

Equation (42) for the parameter $\gamma$ in terms of $\Delta$ takes the form

$$\begin{aligned}&\gamma ^{3}(U^{2}_{1} (0)-U^{2}_{1} (x))[b\beta E_{a} \partial \Psi +(U_{1} (0)+b\partial \Psi )/2]\\&\quad +\gamma ^{2}(U^{2}_{1} (0)-U^{2}_{1} (x))/2-\gamma \Delta (U_{1} (0)+b\partial \Psi )-\Delta =0. \end{aligned}$$

Neglecting the terms with $\gamma ^{3}$, we find a positive solution to the quadratic equation

$$\begin{aligned}&\!\!\gamma \approx [\Delta (U_{1} (0)+b\partial \Psi )\nonumber \\&\quad \!\!+\sqrt{\Delta ^{2}(U_{1} (0)+b\partial \Psi )^{2}+2\Delta (U^{2}_{1} (0)-U^{2}_{1} (x))} ]/(U^{2}_{1} (0)-U^{2}_{1} (x)).\nonumber \\ \end{aligned}$$

(47)

At small values of $\Delta$, when

$$\begin{aligned}&\sqrt{2\Delta (U^{2}_{1} (0)-U^{2}_{1} (x))} \sqrt{1+\Delta (U_{1} (0)+b\partial \Psi )^{2}\frac{1}{2(U^{2}_{1} (0)-U^{2}_{1} (x))}} \\&\quad \approx \sqrt{2\Delta (U^{2}_{1} (0)-U^{2}_{1} (x))}\\&\qquad \times [1+\Delta (U_{1} (0)+b\partial \Psi )^{2}\frac{1}{4(U^{2}_{1} (0)-U^{2}_{1} (x))}+\cdots ] \\ \end{aligned}$$

Equation (47) can be rewritten as

$$\begin{aligned} \gamma\approx & {} \frac{1}{(U^{2}_{1} (0)-U^{2}_{1} (x))}\left[ \sqrt{2\Delta (U^{2}_{1} (0)-U^{2}_{1} (x))}+\Delta (U_{1} (0)+b\partial \Psi )\right. \nonumber \\&\left. \times \left( 1+\frac{\sqrt{2\Delta (U^{2}_{1} (0)-U^{2}_{1} (x))} (U_{1} (0)+b\partial \Psi )}{4(U^{2}_{1} (0)-U^{2}_{1} (x))}+\cdots \right) \right] , \end{aligned}$$

(48)

or

$$\begin{aligned} \gamma= & {} \gamma _{0} \sqrt{\Delta }+\gamma _{1} \Delta +\gamma _{2} \Delta ^{3/2},\quad \gamma _{0} =\sqrt{\frac{2}{(U^{2}_{1} (0)-U^{2}_{1} (x))}} ,\nonumber \\ \gamma _{1}= & {} \frac{U_{1} (0)+b\partial \Psi }{(U^{2}_{1} (0)-U^{2}_{1} (x))},\quad \gamma _{2} =\gamma _{0} (U^{2}_{1} (0)+b^{2}\partial \Psi )^{2}/4. \end{aligned}$$

(49)

We rewrite the condition $\bar{{T}}_{\gamma } >T_{0}$ in the form $T_{0} >T_{0} [1+\gamma (U_{1} (x)+U_{1} (0))+\gamma ^{2}U_{1} (x)U_{1} (0)]$

$$\begin{aligned}&(U_{1} (0)+U_{1} (x))+\gamma U_{1} (0)U_{1} (x)<0,\nonumber \\&\quad \gamma (-U_{1} (x)U_{1} (0))>U_{1} (0)+U_{1} (x). \end{aligned}$$

(50)

This condition can be satisfied for $U_{1} (x)<0,\quad U_{1} (x)+U_{1} (0)>0$, which is possible at certain values of x. Substituting the expression (49) into (50) and neglecting the term with $\Delta ^{3/2}$, we find the condition that $\bar{{T}}_{\gamma } >T_{0}$ in the form

$$\begin{aligned}&\sqrt{\Delta }>\gamma _{+} ,\quad \gamma _{+} =\frac{\sqrt{(U^{2}_{1} (0)-U^{2}_{1} (x))} }{2(U_{1} (0)+b\partial \Psi )}\nonumber \\&\quad \times \left[ \sqrt{1-\frac{2(U_{1} (0)+U_{1} (x))(U_{1} (0)+b\partial \Psi )}{(U_{1} (0)U_{1} (x))}} -1\right] , \end{aligned}$$

(51)

where $\gamma _{+}$ is the positive solution to the quadratic equation

$$\begin{aligned} \gamma _{1} \Delta +\gamma _{0} \sqrt{\Delta }+\frac{U_{1} (0)+U_{1} (x)}{U_{1} (0)U_{1} (x)}=0. \end{aligned}$$

Substitution of (49) into the Eq. (39) for $\bar{{T}}_{\gamma }$ leads to the following formula:

$$\begin{aligned}&\bar{{T}}_{\gamma } =\frac{T_{0} }{R_{2} },\quad R_{2} =1+d_{1/2} \Delta ^{1/2}\nonumber \\&\quad +d_{1} \Delta ^{1}+d_{3/2} \Delta ^{3/2}+d_{2} \Delta ^{2}+d_{5/2} \Delta ^{5/2}+d_{3} \Delta ^{3},\nonumber \\&\quad d_{1/2} =\sqrt{\frac{2(U_{1} (0)+U_{1} (x))}{T_{0} }} ,\quad d_{1} =\frac{1}{T_{0} }\bigg [U_{1} (0)+b\partial \Psi \nonumber \\&\quad +\frac{2}{T_{0} }\frac{U_{1} (0)U_{1} (x)}{(U_{1} (0)+U_{1} (x))^{3/2}}\bigg ],\nonumber \\&\quad d_{3/2} =\sqrt{\frac{2}{T_{0} }} (U_{1} (0)+b\partial \Psi )\bigg [\frac{1}{4}\sqrt{(U_{1} (0)+U_{1} (x))} \nonumber \\&\quad \times (U_{1} (0)+b\partial \Psi )+\frac{2}{T_{0} }\frac{U_{1} (0)U_{1} (x)}{(U_{1} (0)+U_{1} (x))^{3/2}}\bigg ],\nonumber \\&\quad d_{2} =U_{1} (0)U_{1} (x)\frac{\gamma _{0}^{2}}{4}(U_{1} (0)+b\partial \Psi )^{2}(2+\gamma _{0}^{2}),\nonumber \\&\quad d_{5/2} =U_{1} (0)U_{1} (x)\frac{\gamma _{0} ^{2}}{4}(U_{1} (0)+b\partial \Psi )^{2}\gamma _{0} ,\nonumber \\&\quad d_{3} =U_{1} (0)U_{1} (x)\frac{\gamma _{0}^{2}}{4}(U_{1} (0)+b\partial \Psi )^{4}\frac{1}{4}. \end{aligned}$$

(52)

Let us present one more illustration to the above considerations, which, unlike the examples in (18), (23), (38), is based on experimental results. This example clearly shows the importance of taking into account the change in entropy and the effect of external influences on the average FPT.

In [83, 101], the authors studied the fluctuations of the time elapsed until the electric charge transferred through a conductor reached a given threshold value. The distribution of the first-passage times for the net number of electrons transferred between two metallic islands in Coulomb blockade regime is considered. In [83], a simple analytical approximation was derived for the first-passage-time (FPT) distribution, which takes into account the non-Gaussian statistics of the electron transport, and showed that it is capable of describing the experimental distributions with high accuracy.

The average FPT is expressed in terms of the entropy change accompanying this process. In addition to the changes that are necessarily adherent to the course of a random process, any other changes in entropy can be taken into account, which correspond to some other processes in the system. The net effect of such changes affecting the average FPT is illustrated by taking into account different values of the bias voltage. The applied voltage acts as an outer impact on the system. The average FPT with a zero value of the argument of the Laplace transform of the FPT distribution density, which is associated with changes in the entropy of the system, does not reflect the impact of real processes on the average FPT. It is necessary to take into account those changes in entropy that accompany the random process of reaching a certain boundary.

Let us briefly discuss the connection between distribution (8) and superstatistics [8, 9]. There, the assumption is not made about the FPT distribution not depending on the random internal energy u; such a dependence is taken into account. A stochastic storage model [102] is used as a stochastic process for this system. The distributions of superstatistics from [8, 9] can be obtained in another way, different from the approach of [8, 9], if we assume that the intensity of energy flow in the system $\lambda$ depends on the density of the total random internal energy of the system u as $\lambda \rightarrow \lambda _{0} /(n_{0} +u)$. The average value of one stepwise input$x_{0}$ into the system [the input in simple stochastic storage models is a series of Poisson distributed batch inputs with some distribution b(x) of the material in each batch; see [102] for details] is cast as $\bar{{x}}\rightarrow \bar{{x}}_{0} (n_{0} +u)$. For a constant release rate equal to unity, as in the corresponding storage model [96], the partition function $Q_{k}$ does not change; (see for details [8, 9, 96, 102, 103]) $Q_{k}^{-1}=P_{0k} =1-\rho ;\quad \rho =\lambda \bar{{x}},\quad \bar{{x}}=\int {xb(x)\mathrm{{d}}x}$, b(x) is distribution function one stepwise input with average value $\bar{{x}}$. Such substitution can be substantiated by physical considerations that with an increase in the energy of the system and an increase in the number of particles in it, it is more difficult for flows of energy and the number of particles to enter the system.

In our works [8, 9], we have obtained the superstatistics in a way different from that of [104]. This approach is free from the shortcomings pointed out in [105]. The suggested approach can be considered as broadening and detalizing the superstatistics theory. The obtained distribution contains the new parameter related to a thermodynamic state of the system, and also with distribution of a lifetime of a metastable states and interaction of this states with an environment.

The Gibbs distribution does not describe the dissipative processes that develop in the system. Superstatistics describe systems by constantly putting energy into a system with permanent dissipation. The value $\alpha =\gamma /\lambda$ is connected with dissipative processes in the system (through parameter of distribution (8)–(9) $\gamma )$. It defines the correlation between Gibbs and superstatistics multipliers in the distribution from [8, 9]. Other forms of distributions of the form of superstatistics are possible, which can be derived from a distribution of the form (8)–(9) [96].

The original distribution (9) contains two distribution functions: f and R (13), in contrast to superstatistics, which contain only one distribution density for the reciprocal temperature. Therefore, the distribution (9) and distributions obtained from it have greater capabilities than superstatistics. When obtaining expressions from (9) [8,9,10], only the gamma distribution function for $f(T_{\gamma } )$ (13) was used. Another type for this distribution function can be chosen, as well.