First-passage time: a conception leading to superstatistics
To describe the nonequilibrium states of a system we introduce a new thermodynamic parameter – the lifetime (the first passage time) of a system. The statistical distributions that can be obtained out of the mesoscopic description characterizing the behaviour of a system by specifying the stocha...
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Інститут фізики конденсованих систем НАН України
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irk-123456789-1212852017-06-14T03:07:56Z First-passage time: a conception leading to superstatistics Ryazanov, V.V. Shpyrko, S.G. To describe the nonequilibrium states of a system we introduce a new thermodynamic parameter – the lifetime (the first passage time) of a system. The statistical distributions that can be obtained out of the mesoscopic description characterizing the behaviour of a system by specifying the stochastic processes are written. Superstatistics, introduced in [Beck C., Cohen E.G.D., Physica A, 2003, 322A, 267] as fluctuating quantities of intensive thermodynamical parameters, are obtained from statistical distribution with lifetime (random time to system degeneracy) as thermodynamical parameter (and also generalization of superstatistics). 2006 Article First-passage time: a conception leading to superstatistics / V.V. Ryazanov, S.G. Shpyrko // Condensed Matter Physics. — 2006. — Т. 9, № 1(45). — С. 71–80. — Бібліогр.: 37 назв. — англ. 1607-324X PACS: 05.70.Ln, 05.40.-a, 05.10.Gg, 05.20.Gg DOI:10.5488/CMP.9.1.71 http://dspace.nbuv.gov.ua/handle/123456789/121285 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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English |
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To describe the nonequilibrium states of a system we introduce a new thermodynamic parameter – the lifetime
(the first passage time) of a system. The statistical distributions that can be obtained out of the mesoscopic
description characterizing the behaviour of a system by specifying the stochastic processes are written. Superstatistics,
introduced in [Beck C., Cohen E.G.D., Physica A, 2003, 322A, 267] as fluctuating quantities of
intensive thermodynamical parameters, are obtained from statistical distribution with lifetime (random time to
system degeneracy) as thermodynamical parameter (and also generalization of superstatistics). |
| format |
Article |
| author |
Ryazanov, V.V. Shpyrko, S.G. |
| spellingShingle |
Ryazanov, V.V. Shpyrko, S.G. First-passage time: a conception leading to superstatistics Condensed Matter Physics |
| author_facet |
Ryazanov, V.V. Shpyrko, S.G. |
| author_sort |
Ryazanov, V.V. |
| title |
First-passage time: a conception leading to superstatistics |
| title_short |
First-passage time: a conception leading to superstatistics |
| title_full |
First-passage time: a conception leading to superstatistics |
| title_fullStr |
First-passage time: a conception leading to superstatistics |
| title_full_unstemmed |
First-passage time: a conception leading to superstatistics |
| title_sort |
first-passage time: a conception leading to superstatistics |
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Інститут фізики конденсованих систем НАН України |
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2006 |
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http://dspace.nbuv.gov.ua/handle/123456789/121285 |
| citation_txt |
First-passage time: a conception leading to superstatistics / V.V. Ryazanov, S.G. Shpyrko // Condensed Matter Physics. — 2006. — Т. 9, № 1(45). — С. 71–80. — Бібліогр.: 37 назв. — англ. |
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Condensed Matter Physics |
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2025-07-08T19:31:59Z |
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2025-07-08T19:31:59Z |
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1837108427185192960 |
| fulltext |
Condensed Matter Physics 2006, Vol. 9, No 1(45), pp. 71–80
First-passage time: a conception leading to
superstatistics
V.V.Ryazanov∗, S.G.Shpyrko
Institute for Nuclear Research,
Pr. Nauki, 47, 03068, Kiev, Ukraine
Received July 14, 2005
To describe the nonequilibrium states of a system we introduce a new thermodynamic parameter – the lifetime
(the first passage time) of a system. The statistical distributions that can be obtained out of the mesoscopic
description characterizing the behaviour of a system by specifying the stochastic processes are written. Su-
perstatistics, introduced in [Beck C., Cohen E.G.D., Physica A, 2003, 322A, 267] as fluctuating quantities of
intensive thermodynamical parameters, are obtained from statistical distribution with lifetime (random time to
system degeneracy) as thermodynamical parameter (and also generalization of superstatistics).
Key words: stationary states, stochastic processes, noncanonical distributions, slow dynamics and aging
PACS: 05.70.Ln, 05.40.-a, 05.10.Gg, 05.20.Gg
1. Introduction
In [1,2] the generalization of Boltzmann factor exp{−β0E} was introduced in the following
form:
B(E) =
∞
∫
0
dβ′f(β′)e−β′E . (1)
It is supposed there that the intensive parameter (return temperature β, chemical potential, etc.)
fluctuates. These fluctuations evolve on a long time scale. Locally, in some spatial area (cell) where
β is approximately constant, the system is described by usual Boltzmann-Gibbs statistics with
ordinary Boltzmann factor exp{−βE} where E is the energy of a microstate associated with each
cell. On a long time scale it is necessary to take into account fluctuations of β. Superposition of two
statistics (that of β and that of exp{−βE}) which is therefore referred to as “superstatistics” is
derived. This formalism is successfully applied to the description of fully developed hydrodynamic
turbulence, defect motion in convection states, the statistics of cosmic rays and other metastable
and nonequilibrium phenomena. The special case of these superstatistics, at a function f , reduces
to gamma-distribution, and appears in the nonextensive statistical mechanics [3,4] describing a
number of the physical phenomena which are not satisfactorily described by Boltzmann-Gibbs
statistics (for example, long-range many-body systems, systems with memory, numerous phenome-
na in nuclear physics, astrophysics, geophysics, ecology and other complex systems). In the present
work the superstatistics (1) (together with its generalization) is obtained starting with nonequilib-
rium thermodynamics which being a thermodynamic variable contains a first-passage time, lifetime
of statistical system [5–8], section 2.
Stratonovich in [9] used the term “lifetime” as terminus technicus with respect to the number
of phenomena considered. We have encountered the following synonymous expressions: the first
passage time (for some given level), escape time, busy period (in the queuing theory) etc. The first
published paper on the subject is [10] where the Pontryagin equations for the lifetime distribution
are obtained. In 1940 the Kramers work [11] appeared which dealt with the escape time from the
∗E-mail: vryazan@kinr.kiev.ua
c© V.V.Ryazanov, S.G.Shpyrko 71
V.V.Ryazanov, S.G.Shpyrko
potential well. These questions are discussed in books by van Kampen [12], Gardiner [13] and
other authors. The lifetime plays a part in the theory of phase transitions, chemical reactions,
in investigating the dynamics of the complex biomolecules, in calculating the coefficient of the
surface diffusion in semiconductors, in nuclei, elementary particles, spin glasses, spectroscopy, trap
systems, in the theory of metastable states etc.
The lifetime is the time period till the moment of the first (random) reattachment of a certain
level (for example, zero level) by a random process y(t) for the macroscopic parameter (2) (for
example, energy or particle number). The lifetime is the slave process in the terminology of the
random processes theory; that of y(t) determines the behaviour of lifetime. The lifetime depends
on the energy of a system, its size, fluxes of energy. Therefore a system exchanges the energy, the
particle number with the thermostat, but not the “lifetime”. The lifetime Γ is a macroparameter
characterizing the system and its interaction with the environment. It is an observable, well-defined
and physically well understandable quantity reflecting important system peculiarities.
In [14] it is pointed out that a nonequilibrium distribution is characterized by an additional
macroparameter in the description. One can show [15] that in the method of nonequilibrium stati-
stical operator (NSO) [16,17] this parameter is the time t− t0, starting from the birth of a system.
In other words this is the first-passage time of the zero level if we inverse the time, the age of the
system (in Zubarev work [17] the NSO is obtained by averaging over the initial time; in [18] it is
noted that the ε value from NSO in [16,17]is inversely proportional to the lifetime). In works by
Kirkwood [19] it was noted that the state of a system at the current time depends on the whole
evolution history of the nonequilibrium processes in it, hence, on its age as well. Therefore, we
choose the random lifetime as an additional thermodynamical parameter.
It is worth noting that the notion of lifetime has many interpretations, and its physical ap-
plications are also various. For example, the relaxation times can be considered to be lifetimes of
the excited states. An analogy can be traced between the approach of the present work and the
description of nonergodicity in the spin glasses [20] (although the “lifetime” term was not used
there), which included the memory effects (the dependence on the previous history of a system),
the dependence on the initial time moment, and on the times of transition between different valleys
on the energy profile. This problem is related to the response functions in systems with excited
degrees of freedom [21]. In our work we presented the lifetime, like it was done in the pioneering
work [10], from the outset in a strict mathematical form (2), similarly to the same value introduced
in the theory of random processes. One can admit various physical interpretations of the general
mathematical description. One of these possibilities applied in this work to the lifetime of the
system in the energetic valleys, leads to the obtained class of statistical distributions containing
the product of Gibbs-like factor and that of superstatistics.
In the present work we consider the open nonequilibrium systems, stationary nonequilibrium
systems, certain point metastable states. Work on spin glasses and other aging systems, where a
“waiting time” plays an important role, points in this direction.
2. System lifetime and lifetime distribution
The lifetime Γx (or Γ) is thus a random process which is subordinate (in terms of the definitions
of the theory of random processes [22]) with respect to the master process y(t),
Γx = inf{t : y(t) = 0}, y(0) = x > 0. (2)
This definition of the lifetime is taken from the theory of random processes where it is widely
used in the theory of queues, stochastic theory of storage [23]. We use below as y(t) the value of
full energy of the system E, assuming its nonnegativity though it is possible to consider y(t) to be
the number of particles in the system as well.
Gibbs equilibrium distribution for microcanonic probability density in phase space z corre-
sponds to a condition of equiprobability of all possible microstates compatible with the given value
of a macrovariable. We shall assume, that transition of the system in a nonequilibrium conditi-
on breaks the equality of probabilities, characteristically for an equilibrium case. One introduces
72
First-passage time: a conception leading to superstatistics
an additional observable macroparameter, thus extending the phase space (containing additional
degenerate, absorbing states). We shall assume the validity of a principle of equal probabilities
for the extended phase space divided into cells with constant values (E, Γ) (instead of phase cells
with constant values E). The standard procedure (e.g. [24,32]) allows one to write down the relati-
on between the distribution density p(E, Γ) = pEΓ(x, y) and microscopic (coarse-grained) density
ρ(z; E, Γ)
p(E, Γ) =
∫
δ(E − E(z))δ(Γ − Γ(z))ρ(z; E, Γ)dz = ρ(z; E, Γ)ω(E, Γ). (3)
The structure factor ω(E) is thus replaced by ω(E, Γ) – the volume of the hyperspace containing
given values of E and Γ. If µ(E, Γ) is the number of states in the phase space which have the
values of E and Γ less than given numbers, then ω(E, Γ) = d2µ(E, Γ)/dEdΓ. It is evident that
∫ ω(E, Γ = y)dy = ω(E). The number of phase points between E, E + dE; Γ, Γ + dΓ equals
ω(E, Γ)dEdΓ. We make use now of the principle of equiprobability applied to the extended cells
(E, Γ).
Using a maximum-entropy principle [25] (with Shennon’s measure), it is possible to write down
an expression for microscopic (but coarse-grained) density of probability in the extended phase
space
ρ(z; E, Γ) = e−βE−γΓ/Z(β, γ), (4)
where
Z(β, γ) =
∫
e−βE−γΓdz =
∫∫
e−βE−γΓω(E, Γ)dEdΓ (5)
is the partition function, β and γ are Lagrange multipliers satisfying the equations for the averages
〈E〉 = −∂ lnZ/∂β|γ ; 〈Γ〉 = −∂ lnZ/∂γ|β . (6)
Introducing Γ means more information effectively taken into account than merely in linear
terms of the canonical distribution exp{−βE}/Z(β). The value γ thermodynamically conjugated
to the lifetime is related to the entropy fluxes and entropy production which characterize the
peculiarities of the nonequilibrium processes in an open thermodynamic system. If γ = 0 and
β = β0 = (kBTeq)
−1, where kB is the Boltzmann constant, Teq is the equilibrium temperature,
then the expressions (4), (5) yield the equilibrium Gibbs distribution. One can thus consider (4)–
(5) as a generalization of the Gibbs statistics to cover the nonequilibrium situation. The physical
phenomena such as metastability, phase transitions, stationary nonequilibrium states are known
to violate the equiprobability of the phase space points. The value γ can be regarded as a measure
of deviation from the equiprobability hypothesis. The value of γ = 0 characterizes equilibrium
isolated systems, presenting an idealization. If the detailed balance is satisfied, then in equilibrium
γ 6= 0. Mathematically introducing lifetime means acquiring additional information regarding the
underlying stochastic process, beyond merely the knowledge of its stationary distribution, exploring
the (stationary) properties of its slave process.
Let us underline the principal features of the suggested approach.
1. We introduce a novel variable Γ which can be used to derive additional information regarding
the system in a stationary nonequilibrium state. We suppose that Γ is a measurable quantity
at macroscopic the level, thus values like entropy which are related to the order parameter
(principal macroscopic variable) can be defined. At the mesoscopic level the variable Γ is
introduced as a variable with operational characteristics of a random process slave with
respect to the process describing the order parameter.
2. We suppose that thermodynamic forces γ related to the novel variable can be defined. One can
introduce the “equations of state” β(〈E〉, 〈Γ〉), γ(〈E〉, 〈Γ〉). Thus we introduce the mapping
(at least approximate) of the external restrictions on the point in the plane β, γ.
3. We suppose that a “refined” structure factor ω(E, Γ) can be introduced which satisfies the
condition ∫ ω(E, Γ = y)dy = ω(E) (ordinary structure factor). This function (like ω(E))
73
V.V.Ryazanov, S.G.Shpyrko
is the internal (inherent) property of a system. At the mesoscopic level we can ascribe to
this function some random process inherent to the system (at given restrictions (β0, γ0)).
The structure factor has the meaning of the joint probability density for the values E, Γ
understood as the stationary distribution of this process.
4. It is supposed that at least for certain classes of effects the resulting distribution has the form
(4), (5), i.e., the change of the principal random process belongs to some class of invariance
leading to this distribution which explains how one can pass from the process in the reper
point (β0, γ0) (for example, in equilibrium when γ = 0 and β = 1/kBT ) to a system in an
arbitrary nonequilibrium stationary state. The thermodynamic forces should be chosen so
that the distribution leads to new (measurable) values of (〈E〉, 〈Γ〉).
3. Distribution for indisturbed lifetime
The use of a specific lifetime distribution depends on the physical peculiarities of the system
under investigation. Further detailed elaboration will require the concrete definition of the Γ dis-
tribution and the interpretation of the Lagrange parameters β and γ. The Lagrange parameter β
is supposed to be (like the equilibrium Gibbs statistics) as follows:
β = 1/kBT, (7)
where T is the average (over the body volume) local equilibrium temperature. Since at fluxes ~q 6=0
the temperature is not the same all over the bulk of the body, one can define T in a system with
volume V as the volume average, i.e.
T = V −1
∫
V
T (r, t)dr,
(the same definition was used in [24]). Of course, the thermodynamic description itself is supposed
to be already coarse-grained. To get the explicit form of the Γ distribution we shall use the general
results of the mathematical theory of phase coarsening of the complex systems [26], which imply
the following density of distribution of the lifetime for coarsened random process (see also [9]):
p(Γ < y) = Γ−1
0 exp(−y/Γ0) (8)
for one class of the stable states and the Erlang density
p(Γ < y) =
n
∑
k=1
RkΓ−1
0k exp(−y/Γ0k);
n
∑
k=1
Rk = 1 (9)
in the case of several (n) classes of the ergodic stable states, Rk – probability that the system is
in the k-th class. The values Γ0 and Γ0i are averaging of the residence times and the degeneracy
probabilities over stationary ergodic distributions (in our case – Gibbs distributions). The physical
reason for the realization of the distribution in the form (8), (9) is the existence of the weak
ergodicity in a system. Mixing the system states at big times will lead to the distributions (8), (9).
As we note in Section 2, the structure factor ω(E, Γ) has a meaning of the joint probability density
of values E, Γ. For the distributions (9), (10) the functions ω(E, Γ) from (3), (5) take on the form:
ω(E, Γ) = ω(E)Γ−1
0 exp(−Γ/Γ0); ω(E, Γ) = ω(E)
n
∑
k=1
RkΓ−1
0k exp(−Γ/Γ0k) (10)
for the relations (9) and (10) respectively. Substituting into the partition function yields
Z(β, γ) = Z(β)(1 + γΓ0)
−1; Z(β, γ) = Z(β)
n
∑
k=1
Rk(1 + γΓ0k)−1 (11)
74
First-passage time: a conception leading to superstatistics
for (8) and (9), where Z(β) = ∫ ω(E) exp(−βE)dE is the Gibbs partition function.
Let us note that the value similar to γ is defined in [27–29] for fractal objects. It is equal to
zero for the closed system, and for open system it is equal to Σλi − λKS, where λi are Lyapunov’s
parameters, and λKS is Kolmogorov-Sinai entropy.
4. Superstatistics from distribution of the kind (5)
In the distribution (4), (5) containing lifetime, as thermodynamic parameter, joint probability
for values E and Γ is equal to
p(E, Γ) = e−βE−γΓω(E, Γ)/Z(β, γ);
Z(β, γ) =
∫
e−βE−γΓdz =
∫∫
e−βE−γΓω(E, Γ)dEdΓ. (12)
Having integrated (14) on Γ, we obtain distribution of a kind
p(E) =
∫
p(E, Γ)dΓ = exp(−βE)
∫
∞
0
exp(−γΓ)ω(E, Γ)dΓ/Z(β, γ). (13)
We shall write down
ω(E, Γ) = ω(E)ω1(E, Γ) = ω(E)
n
∑
k=1
Rkfk(Γ, E). (14)
In the last equality (14) it is supposed that there exist n classes of ergodic states in a system;
Rk is the probability that the system will be in k-th class of ergodic states, fk(Γ, E) is density of
distribution of lifetime Γ in this class of ergodic states (generally fk depends on E). As physical
example of such situation (characteristic of metals, glasses) one can mention the potential of many
complex systems. Such a situation is considered in [30]. Minima of potential correspond to the
metastable phases, disproportionate structures, etc. Expression (14) is applicable to the description
of such systems.
We shall assume an obvious kind of distribution fk in (14), having chosen it as gamma-
distribution
fk(x) = xαk−1e−x/bk/Γ(αk)bαk
k , x > 0, fk(x) = 0, x < 0;
∞
∫
0
e−γkxfk(x)dx = (1 + γkbk)−αk , (15)
(Γ(α) is gamma-function). We assume that in k-th metastable area γ = γk. We shall choose
αk = γk/λk; bk = Γ0kλk/γk, λk is the intensity of energy flow in the system (subsystem), equal
in dynamical equilibrium of an output intensity [31]. For realistic metastable systems the detailed
balance principle is satisfied and λ 6= 0. Then bkαk = Γ0k, (1 + γkbk)−αk = (1 +λkΓ0k)−γk/λk ; Γ0k
is equal to average lifetime in k-th class of metastable states without disturbance from (8)–(11).
We understand the absence of stationary forces and the determined fluxes as undisturbed lifetime
Γ0k; at the system (open metastable area) there are only stochastic fluxes. For average lifetime Γ0k
of the system in dynamical equilibrium (or in stationary) state, where at the system with grand
canonical sum Qk there are only stochastic fluxes, in work [31] by means of stochastic models of
storage it is obtained
Γ0k = (Qk − 1)/λk; 1 + λkΓ0k = Qk, (16)
where Qk = exp{βkPkVk} is the grand statistical sum of the grand canonical ensemble of the part
of system in k-th metastable state, in the k-th potential well, βk = (1/kBT )k is reverse temperature
in k-th metastable state, Pk, Vk, Tk are pressure, volume, temperature in k-th metastable state, V
is the full system volume.
75
V.V.Ryazanov, S.G.Shpyrko
From (15)–(16) it is seen that
∞
∫
0
e−γxfk(x)dx = (1 + λkΓ0k)−αk = Q−αk
k = e−αk(P/kBT )kVk ; αk = γk/λk. (17)
For exponential distribution fk in (15) αk = 1. Substitution (14), (17) in (13) gives
p(E) = e−βEω(E)Z−1
n
∑
k=1
Rke−βk(Pkvk/u)αkE , (18)
where u = E/V is specific energy, vk = Vk/V . At such an approach which can be considered
as a discretized version of superstatistics [1,2], one accounts not only for the inverse temperature
fluctuations (that of β), but also for the fluctuations of values P, v, α. Denote in (18) (Pkvkαk)/u =
rk and rewrite the exponent in (18) as ykβ0r0E, where β0 = ΣRkβk ≈ Σβk/n, r0 = ΣRkrk ≈ Σrk/n
are averages of βk and rk; yk = (βkrk/β0r0). It will be shown below that r0 is related to the
controlling parameter of a problem (for example, with the feedback coefficient in the Van der Pol
generator [24], the birth parameter in the Maltus-Ferhuelst process etc). Since Rk is the probability
that the system will be in k-th class of ergodic states, then Rk is the probability that the quantity
y = (βr/β0r0) will take on the value yk = (βkrk/β0r0); Rk = R(y = yk). Thus the expression
under sum in r.h.s. of (19) is the characteristic function ϕ(t) =
∑n
i=1 eitxiRi of the distribution Rk
with the argument it = −β0r0E. If Rk = δk0, where for a cell with zero index βkrk = β0r0, then
(18) acquires the form p(E) ∼ exp(−βE − β0r0E) = exp(−β0E), where the open character of the
system (the existence of fluxes) is taken into account, therefore the inverse temperature is set to
be not β = 1/kT , but β0. Thus the relation between β and β0 is β = β0(1 − r0); β0 = β/(1 − r0).
We shall write down (18) in the form
p(E) = e−βEω(E)ϕ(E)/ZA; ϕ(t) =
n
∑
i=1
eitxiRi; it = −β0r0E (19)
and consider different resources of the task of function ϕ(E) in (19). Let us choose as Rk in (19) the
probabilities of the negative binominal distribution (a discrete analogue of the continuous gamma
distribution) with a characteristic function ϕ(t) =
∑n
i=1 eitxiRi = [p/(1 − (1 − p) exp(it))]r, r and
p being parameters. Setting r = 1/(q − 1), it = −β0r0E, p = 1/[1 + (q − 1)] = 1/q, we conclude
that (19) in this case is
p(E) ∼ e−βE
[
1 + (q − 1)(1 − e−β0r0E)
]−1/(q−1)
. (20)
In this approach the behaviour of p(E) is asymptotically not power-like, as in Tsallis [3,4] dis-
tributions, but rather exponential. The argument β0r0E in the Tsallis distribution [3,4] is replaced
by (1 − e−β0r0E). As Rk one can choose various specific forms of distributions.
As example of application of expressions (4)–(6), (12)–(14) can consider a flow of rarefied gas
in a tube of radius R and lengths L, L � R. By use of (8)–(11), (15) and storage model [31] we
obtain
p(E) = e−βEω(E)/Z[1 − 2R(1/2m)1/2(n1 − n2)(kBE/CV)1/2/3πLaf ],
where E is microscopic random internal energy, m is weight of atoms, n1, n2 are densities of gas on
the ends of a tube, CV is a thermal capacity of system at a constant volume, value af=const can
be defined as the maximal value of a flux, at which there exist stationary states in laminar mode,
flux in a tube is q = 2R(m/2kBT )1/2(P1 − P2)/3πL = 2R(1/2m)1/2(n1 − n2)(kBU/CV)1/2/3πL
[32]. Value q = af is a point of nonequilibrium phase transition from a laminar mode to turbulent.
As another example we shall consider the system of neutrons in a nuclear reactor and we shall
obtain distribution of energy of neutrons in a nuclear reactor in view of finiteness of their lifetime.
76
First-passage time: a conception leading to superstatistics
5. Continuos approximation for the superstatistics factor
If in expressions (18)–(19) we denoted r1 = P ′v′α′/u, αk → α′(x), Rk → R′(x), f(x) =
dR′(x)/dx; vk → v′(x), Tk → T ′(x), β1 = 1/kBT ′(x), and proceeding to continuous sampling space,
from discrete probabilities to continuous probability density function, and replaced summation by
integration and defining density of probability
f(β1r1/β0r0) =
dR′(x)/dx
d(β1r1/β0r0)/dx
,
we obtain from (19):
p(E) = e−βEω(E)Z−1
∞
∫
0
f(y1)e
−y1β0r0Ed(y1);
∞
∫
0
f(y)dy = 1. (21)
Here we assumed that f(β1r1/β0r0) is the probability density for a random value y(x) = β(x)r(x)/β0r0
to take on a specific value y1 = β1r1/β0r0 under the condition that the system is located at the
point x of the potentially possible states considered as a continuous random value (although the
initial set of potential values is a discrete set). Like in (18) we wrote (Pkvkαk)/u = rk and rep-
resented the exponent in (21) as −y1β0r0E, y1 = (β1r1/β0r0), taking β0r0E the argument of the
characteristic function. Then the integrand in the r.h.s. of (21) is the Laplas transform for the
distribution f ′(y1) with argument β0r0E.
In the open system with complex potential an extra factor αvβ appears, taking into account
the exchange with surrounding (value α) and the fluctuations of the inverse temperature β and
values of P , v, α. If one performs the replacement of variables β′ = β + r1β1 and assumes that
the situation β = 0 is possible, when the bottom limit of integration in (21) after replacement of
a variable is 0, instead of (21) we shall obtain
p(E) = ω(E)Z−1
∞
∫
0
f1(β
′) exp{−β′E}dβ′,
f1(β
′) = f(β1r1) = f [β′ − β0(1 − r0)],
which coincides with superstatistics [1]. But the correlation (21) describes a more general situation
and superstatistics forms here a special case (21). The value
r0 = α0(P0v0/u); r0 = 〈r〉 =
∫
rf(r)dr; r0 = 〈αPv/u〉; α = γ/λ,
α0 =
∫
α1f(β1r1)d(β1r1), r = α/u, P0 = 〈P 〉 =
∫
P1f(β1r1)d(β1r1).
For ideal gas P/u ≈ 0, 687. It is possible to mention some examples of physical realization of
distribution (21). So, for a case of superdiffusion in [33] at the absence of external force, stati-
onary distribution is in Tsallis form, and for a case of the presence of constant external force,
the multiplicative noise in exponential form of distribution is manifested. As in (21), two kinds of
distributions are combined in a single phenomenon for one system. The obtained distribution of
the kind (21) provides ample opportunities for describing various physical situations.
In [34] it is shown that the temperature fluctuations can be described using the gamma-
distribution. As in [1] for distribution function f in (21) similar to the gamma-distribution in (15)
(but with variables αk = c, bk = b; f(y) = Γ−1(c)b−c(y)c−1 exp{−y/b}) expression (21) reduces to
the product of Gibbs distribution and Tsallis distribution of a kind (c = 1/(q − 1); b = q − 1)
p(E) = e−βEω(E)Z−1(1 + bE)−c = e−βEω(E)Z−1[1 + (q − 1)β0r0E]−1/(q−1). (22)
At q → 1 one gets from (22) the expression exp{−β0E}, which at r0 = 0 yields Gibbs distribution
exp{−βE}. In (22) we can separate the Tsallis temperature, βq = βTs = βr0/(1−r0) = β0r0. Thus
77
V.V.Ryazanov, S.G.Shpyrko
we do have three different “temperatures”: Gibbs temperature (7) β = βGibbs = 1/kBT = β0(1−r0);
effective temperature β0 = βeff = β/(1− r0); and Tsallis temperature βq = βTs = β0r0 = βr0/(1−
r0). The following relations between them hold: β0 = β + βq; r0 = 1 − β/β0 = βq/β0. It is obvious
that distribution (21) represents the product of Boltzmann factor with a multiplier superstatistics
from which Tsallis distribution as a special case can be obtained (having chosen f in (21) the
gamma-distribution of a kind (15)). The behaviour of distribution (21), (22) essentially depends
on the values r0 (α0). At r0 → 0 (α0 → 0, α0 > 0) the Gibbs distribution is obtained. At α → 0
and γ → 0. As it is marked in the beginning of section 3, it corresponds to transition to Gibbs
distributions. Thus, not only λ → 0, but also α = γ/λ → 0. The value γ describes entropy fluxes
and entropy production in the system [15]; the value λ describes the intensity of an input in the
system (and an output from it). Similarly to the discrete case, one can specify the expressions
obtained. As f in (21) one can choose various continuous expressions.
The distributions (21), (22) are widely encountered in various domains of physics. For example,
for distributions in the systems with multiplicative noise in [35] the stationary solution is written
as
p(u) = C/η(u); C = const; η(u) = (Mg2 + A)1/2e−
�
h(u)du/(Mg2+A), (23)
where du/dt = h(u) + g(u)ξ(t) + η(t), u(t) is a stochastic variable, h, g are arbitrary functions
(g(0) = 0), and ξ(t), η(t) are uncorrelated and Gaussian-distributed zero-mean white noises, hence
satisfying 〈ξ(t)ξ(t′)〉 = 2Mδ(t − t′), 〈η(t)η(t′)〉 = 2Aδ(t − t′), where M, A > 0 are the noise
amplitudes and stand for multiplicative and additive, respectively. If we assume the deterministic
drift h(u) and g(u) in the form widely used in synergetics (for example, [36]): h(u) = αu −
δu3; g(u) = u, then from (23):
p(u) ∼ e−δu2/2M/[1 + Mu2/A][1−(α+δA/M)/M ]/2. (24)
Comparing (24) and (22), we get
E = u2/2; δ/M = β = 1/kBT ; 1/(q − 1) = [1 − (α + δA/M)/M ]/2;
(q − 1)β0r0 = M/A; r0 = 1 − δA/M(M − α) = 1 − βA/(M − α);
β0 = β/(1 − r0) = β + βq = (M − α)/A;
βq = β0r0 = (M − α)/A − β = β0 − β.
Hence, it is clear that the sign of q − 1, r0 depends on the relation between α, δ, A, M . The value
β0 = β at α = M − βA. The value q − 1 has a singularity at 1 = (α + δA/M)/M , changing its
sign. At α = M the value β0 turns to zero and the value r0 has a singular behaviour. It follows
that the value of the controlling parameter α = M corresponds to the bifurcation point. Besides
the (24) one can name a number of different examples of choosing functions h(u) and g(u) [30,35].
Not only gamma-distributions (yielding Tsallis distribution (22)), but also other possible classes
of functions can be used as f in (21).
Turbulence, Van der Pol generators, the velocity distribution in the presence of the heat source,
space diffusion of the Brownian particles in the external field, Maltus-Ferhuelst process, Van der
Pol-Duffing system [24] and many examples of distributions of other phenomena (self-organized
criticality, a spectrum of the cosmic ray) which lead to expressions of the kind (21), (22) are
described in the same way.
6. Conclusion
From the distribution (5) containing lifetime of statistical system (first-passage time, time
before degeneration of system) we obtained the energy distributions of a system which contain the
product of the Gibbs factor and a factor corresponding to the superstatistics. In the superstatistics
factor one averages not only over the values of the finite temperature, but also over other system
parameters, including the parameters of interaction with the environment.
The superstatistics theory [1,2] is the generalization of the nonextensive statistical mechanics
[3,4]. Some of the fields of applications of the nonextensive statistical mechanics are metastable
78
First-passage time: a conception leading to superstatistics
states, Levy flights, systems of finite size [37]. All these phenomena are conditioned physically by
the finiteness of the lifetime, and by the possibilities of small probability big fluctuations which
are generally dismissed when accounting for the system evolution.
The introduction of the lifetime (first-passage time) as a thermodynamic parameter is justified
by the fact that finite systems possess finite lifetimes, which essentially effects their properties and
the properties of their environment. The lifetime is interpreted as a fundamental quality having
dual nature related to both the flux of external time, and the properties of a system. In [15] the
expressions for the lifetime through dynamical characteristics of a system were obtained. These
expressions show the dependence on the interchange with the environment.
The Gibbs distribution does not describe the dissipative processes that develop in the system.
Superstatistics describe systems by constantly putting the energy into the system, which is dissi-
pated. The value α = γ/λ is related to dissipative processes in the system (through γ). It defines
a correlation between Gibbs and superstatistics multipliers in the distribution (21). The value γ
thermodynamically conjugated to random lifetime is expressed through entropy production and
currents [7,15], i.e. through communication of the system with an the environment. The obtained
distribution contains a new parameter related to a thermodynamic state of the system, as well
as with distribution of a lifetime of metastable states and interaction of these states with the
environment. Changing this parameter it is possible to go over to Gibbs distribution.
Thus, the lifetime finiteness being explicitly taken into account for a statistical system and
the use of the lifetime (first-passage time) as a thermodynamic variable brings us to the nonex-
tensive statistical mechanics. This fact is in correspondence with the known results regarding the
finiteness of the size for systems in which the nonextensive statistical mechanics can be applied
[37]. Furthermore we obtained the distribution generalizing the Gibbs-Boltzmann statistics and
superstatistics. Real open systems seem to share the properties of both Gibbs and nonextensive
statistical mechanics. So we believe such kinds of distributions to be a promissing point of view,
which is also confirmed by the relevance of such distributions to various phenomena. Depending on
the value of r0 and on the controlling parameter, the properties of either nonextensive or extensive
statistical mechanics prevail. The obtained relation (21) can be regarded as a detailed elaboration
of the superstatistics theory [1]. For the expression (21) one can write the entropy expression as
well as the corresponding thermodynamic relations.
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80
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