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Response of a thermodynamic system subject to stochastic thermal perturbations
Physics Letters A 379 (2015) 3035–3036
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Response of a thermodynamic system subject to stochastic thermal perturbations V. Bertola a , E. Cafaro b a b
School of Engineering, University of Liverpool, Brownlow Hill, Liverpool L69 3GH, United Kingdom Dipartimento di Energetica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
a r t i c l e
i n f o
Article history: Received 15 September 2015 Accepted 9 October 2015 Available online 20 October 2015 Communicated by F. Porcelli Keywords: Thermodynamic system Random perturbations Fokker–Planck equation
a b s t r a c t The thermal response of a closed thermodynamic system perturbed by random fluctuations of the heat transfer rate through its boundary is studied by means of stochastic differential equations, in analogy with the mesoscopic thermodynamics approach. It is shown that the probability density of the thermodynamic beta (inverse temperature), which reflects the system randomisation due to the perturbation, can be significantly different from that of the perturbation itself. © 2015 Elsevier B.V. All rights reserved.
Equilibrium thermodynamics provides a characterisation of physical systems in terms of a limited number of variables accounting for average properties, neglecting the disordered motion of large numbers of molecules. However, in many systems of practical interest the number of elementary particles is large enough to neglect the behaviour of individual particles, but at the same not enough large to neglect fluctuations completely as in equilibrium thermodynamics [1]. These systems, characterised by a number of elementary particles smaller than Avogadro’s number (1 N N AV ) are relevant to applications including transport phenomena in materials and in biology, chemical and biochemical kinetics, adsorption, thermoionic emission, spin flip processes, and many others. Such mesoscopic systems cannot be described by linear irreversible thermodynamics, which does not account for fluctuations, but require a different approach [2–4], where the mesoscopic dynamics of the system is described in probabilistic terms through the Fokker–Planck equation relative to a diffusion process of the probability compatible with the statistical formulation of the second law of thermodynamics [5]. This approach, known as mesescopic thermodynamics [6,7], was successfully applied to the study of adsorption [8], chemical reactions [9], nucleation [10], active transport in ion channels [11], molecular motors [12], evaporation and condensation [13]. A similar approach can be used to investigate thermodynamic systems, which although initially at equilibrium, are subjected to
E-mail address: [email protected] (V. Bertola). http://dx.doi.org/10.1016/j.physleta.2015.10.026 0375-9601/© 2015 Elsevier B.V. All rights reserved.
external random perturbations. The internal response of the system to such perturbations, i.e. its internal randomization, can be quantified by the thermodynamic beta (or thermodynamic perk) of the system, which is usually calculated in the microcanonical ensemble as the reciprocal of the thermodynamic temperature of a system, β = 1/k B T . It provides the connection between the theoretic/statistical interpretation of a physical system through its information entropy and the thermodynamics associated with its energy, and expresses the response of entropy to an increase in energy. If a system is challenged with a small amount of energy, then β describes the amount by which the system will randomise [14]. Here, we investigate the response of a closed thermodynamic system, i.e. exchanging energy but not mass with its surrounding, which experiences a Gaussian energy flow through its boundary. In particular, we consider a small sub-system, which is part of a large thermodynamic system at equilibrium. The overall energy density satisfies the following conservation equation:
∂ E (x, t ) + ∇ · J(x, t ) = 0 ∂t
(1)
with
J(x, t ) = J D (x, t ) + J S (x, t )
(2)
where J D (x, t ) and J S (x, t ) are, respectively, the deterministic and the stochastic components of the energy density flow.
3036
V. Bertola, E. Cafaro / Physics Letters A 379 (2015) 3035–3036
Integrating Eq. (1) with respect to volume and applying the Gauss formula, one obtains the energy conservation equation for the sub-system:
where C = C τ 2 h/2k B ; after integration with respect to temperature one obtains
dE
P (T ) =
dt
+ Q˙ D (t ) + Q˙ S (t ) = 0
with
Q˙ D (t ) =
(3)
J D , S (x, t ) · nd
(4)
where is the surface area of the sub-system boundary, and Q˙ S (t ) is the stochastic component of the heat transfer rate through the sub-system boundary. The internal energy and the deterministic part of the energy flow are given by the following constitutive equations:
E (t ) = c V T (t )
Q˙ D (t ) = h T (t ) − T 0
(5)
(6)
where c v is the heat capacity of the sub-system, h is the heat transfer coefficient, and T 0 is the average temperature of the system. With these constitutive equations, Eq. (3) takes the form:
cV
dT dt
= −h T (t ) − T D
− Q˙ S (t )
ξ(t ) = 0 ξ(t )ξ t = 2k B hδ t − t
dt
=−
1
τ
T (t ) − T 0 −
ξ(t )
(11)
where τ = c V /h . The corresponding Fokker–Planck equation for the temperature distribution function is:
1 ∂2 ∂ ∂ P (T , t ) =− A(T ) P (T , t ) + B (T ) P (T , t ) 2 ∂t ∂T 2 ∂T
(12)
with
1 A(T ) = − (T − T 0 ) +
τ
B (T ) =
2k B
τ 2 h
τ
kB 2 h
T
T2
P (T ) =
B (T )
T exp 2
A (η) B (η)
P (T ) =
T
exp −
τ h kB
T 0
η − T0 dη η2
C
+∞ = β γ −1 exp(−γ k B T 0 β)dβ = 0
1
(γ k B T 0 )γ
(γ )
(19)
Thus, the probability distribution of the normalised inverse temperature becomes
(γ k B T 0 )γ γ −1 exp(−γ k B T 0 β) β
(γ )
1
(γ )
zγ −1 exp(− z)
(20)
(21)
Eq. (21) shows that when a thermodynamic system is perturbed with a Gaussian stochastic energy supply thorough its boundary, the internal randomisation of the system (i.e., its entropy fluctuations) is not Gaussian, but is described by a gamma probability density function. This conclusion, which is not evident a priori, demonstrates that the system randomisation due to external perturbations does not necessarily have the same characteristics of the perturbation itself. More in general, this work shows that the approach used in mesoscopic thermodynamics can be extended to thermodynamic systems which do not strictly belong to the category of mesoscopic systems whose internal perturbations are not negligible, but where perturbations are introduced from outside.
[1] R.S. Ingarden, Towards mesoscopic thermodynamics: small systems in higherorder states, Open Syst. Inf. Dyn. 1 (1992) 75. [2] J.M.G. Vilar, J.M. Rubi, Proc. Natl. Acad. Sci. USA 98 (2001) 11081. [3] D. Reguera, J.M. Rubi, J.M.G. Vilar, J. Phys. Chem. B 109 (2005) 21502. [4] J.M. Rubi, Sci. Am. 299 (2008) 62. [5] A. Perez-Madrid, J.M. Rubi, P. Mazur, Physica A 212 (1994) 231. [6] I. Santamaria-Holek, J.M. Rubi, A. Perez-Madrid, New J. Phys. 7 (2005) 35. [7] J.M. Rubi, Mesoscopic thermodynamics, Phys. Scr. T 151 (2012) 014027. [8] I. Pagonabarraga, J.M. Rubi, Physica A 188 (1992) 553. [9] I. Pagonabarraga, A. Perez-Madrid, J.M. Rubi, Physica A 237 (1997) 205. [10] D. Reguera, J.M. Rubi, J. Chem. Phys. 109 (1998) 5987. [11] S. Kjelstrup, J.M. Rubi, D. Bedeaux, J. Theor. Biol. 234 (2005) 7. [12] S. Kjelstrup, J.M. Rubi, D. Bedeaux, Phys. Chem. Chem. Phys. 23 (2005) 4009. [13] D. Bedeaux, S. Kjelstrup, J.M. Rubi, J. Chem. Phys. 119 (2003) 9163. [14] C. Kittel, H. Kroemer, Thermal Physics, 2nd ed., W.H. Freeman and Company, United States of America, 1980.
(15)
where the integration constant C is determined by the normalisation condition. Substituting Eqs. (13) and (14), Eq. (15) becomes:
1
(14)
0
C
γ
where C = C k B ; the integration constant is given by:
References
dη
(18)
(13)
The coefficients A and B are not explicitly dependent on time. The Fokker–Planck equation has a steady-state solution given by:
C
P (β) = C β γ −1 exp(−γ k B T 0 β)
P ( z) = (10)
cV
where γ = c V /k B . The probability density function of temperature can be expressed in terms of the normalised inverse temperature β = 1/k B T , and recalling that dβ = −(1/k B T 2 )dT yields:
and with the further variable change z = γ k B T 0 β yields a gamma distribution:
(9)
T (t )
(17)
(8)
and k B is Boltzmann’s constant. Under these assumptions, the subsystem temperature is described by the following Langevin equation:
dT (t )
(7)
where ξ(t ) is a Gaussian stochastic process with
T0 exp −γ 1 + γ T T
P (β) =
We define the stochastic component of the heat rate as:
Q˙ S (t ) = T (t )ξ(t )
C
(16)
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Response of a thermodynamic system subject to stochastic thermal perturbations V. Bertola a , E. Cafaro b a b
School of Engineering, University of Liverpool, Brownlow Hill, Liverpool L69 3GH, United Kingdom Dipartimento di Energetica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
a r t i c l e
i n f o
Article history: Received 15 September 2015 Accepted 9 October 2015 Available online 20 October 2015 Communicated by F. Porcelli Keywords: Thermodynamic system Random perturbations Fokker–Planck equation
a b s t r a c t The thermal response of a closed thermodynamic system perturbed by random fluctuations of the heat transfer rate through its boundary is studied by means of stochastic differential equations, in analogy with the mesoscopic thermodynamics approach. It is shown that the probability density of the thermodynamic beta (inverse temperature), which reflects the system randomisation due to the perturbation, can be significantly different from that of the perturbation itself. © 2015 Elsevier B.V. All rights reserved.
Equilibrium thermodynamics provides a characterisation of physical systems in terms of a limited number of variables accounting for average properties, neglecting the disordered motion of large numbers of molecules. However, in many systems of practical interest the number of elementary particles is large enough to neglect the behaviour of individual particles, but at the same not enough large to neglect fluctuations completely as in equilibrium thermodynamics [1]. These systems, characterised by a number of elementary particles smaller than Avogadro’s number (1 N N AV ) are relevant to applications including transport phenomena in materials and in biology, chemical and biochemical kinetics, adsorption, thermoionic emission, spin flip processes, and many others. Such mesoscopic systems cannot be described by linear irreversible thermodynamics, which does not account for fluctuations, but require a different approach [2–4], where the mesoscopic dynamics of the system is described in probabilistic terms through the Fokker–Planck equation relative to a diffusion process of the probability compatible with the statistical formulation of the second law of thermodynamics [5]. This approach, known as mesescopic thermodynamics [6,7], was successfully applied to the study of adsorption [8], chemical reactions [9], nucleation [10], active transport in ion channels [11], molecular motors [12], evaporation and condensation [13]. A similar approach can be used to investigate thermodynamic systems, which although initially at equilibrium, are subjected to
E-mail address: [email protected] (V. Bertola). http://dx.doi.org/10.1016/j.physleta.2015.10.026 0375-9601/© 2015 Elsevier B.V. All rights reserved.
external random perturbations. The internal response of the system to such perturbations, i.e. its internal randomization, can be quantified by the thermodynamic beta (or thermodynamic perk) of the system, which is usually calculated in the microcanonical ensemble as the reciprocal of the thermodynamic temperature of a system, β = 1/k B T . It provides the connection between the theoretic/statistical interpretation of a physical system through its information entropy and the thermodynamics associated with its energy, and expresses the response of entropy to an increase in energy. If a system is challenged with a small amount of energy, then β describes the amount by which the system will randomise [14]. Here, we investigate the response of a closed thermodynamic system, i.e. exchanging energy but not mass with its surrounding, which experiences a Gaussian energy flow through its boundary. In particular, we consider a small sub-system, which is part of a large thermodynamic system at equilibrium. The overall energy density satisfies the following conservation equation:
∂ E (x, t ) + ∇ · J(x, t ) = 0 ∂t
(1)
with
J(x, t ) = J D (x, t ) + J S (x, t )
(2)
where J D (x, t ) and J S (x, t ) are, respectively, the deterministic and the stochastic components of the energy density flow.
3036
V. Bertola, E. Cafaro / Physics Letters A 379 (2015) 3035–3036
Integrating Eq. (1) with respect to volume and applying the Gauss formula, one obtains the energy conservation equation for the sub-system:
where C = C τ 2 h/2k B ; after integration with respect to temperature one obtains
dE
P (T ) =
dt
+ Q˙ D (t ) + Q˙ S (t ) = 0
with
Q˙ D (t ) =
(3)
J D , S (x, t ) · nd
(4)
where is the surface area of the sub-system boundary, and Q˙ S (t ) is the stochastic component of the heat transfer rate through the sub-system boundary. The internal energy and the deterministic part of the energy flow are given by the following constitutive equations:
E (t ) = c V T (t )
Q˙ D (t ) = h T (t ) − T 0
(5)
(6)
where c v is the heat capacity of the sub-system, h is the heat transfer coefficient, and T 0 is the average temperature of the system. With these constitutive equations, Eq. (3) takes the form:
cV
dT dt
= −h T (t ) − T D
− Q˙ S (t )
ξ(t ) = 0 ξ(t )ξ t = 2k B hδ t − t
dt
=−
1
τ
T (t ) − T 0 −
ξ(t )
(11)
where τ = c V /h . The corresponding Fokker–Planck equation for the temperature distribution function is:
1 ∂2 ∂ ∂ P (T , t ) =− A(T ) P (T , t ) + B (T ) P (T , t ) 2 ∂t ∂T 2 ∂T
(12)
with
1 A(T ) = − (T − T 0 ) +
τ
B (T ) =
2k B
τ 2 h
τ
kB 2 h
T
T2
P (T ) =
B (T )
T exp 2
A (η) B (η)
P (T ) =
T
exp −
τ h kB
T 0
η − T0 dη η2
C
+∞ = β γ −1 exp(−γ k B T 0 β)dβ = 0
1
(γ k B T 0 )γ
(γ )
(19)
Thus, the probability distribution of the normalised inverse temperature becomes
(γ k B T 0 )γ γ −1 exp(−γ k B T 0 β) β
(γ )
1
(γ )
zγ −1 exp(− z)
(20)
(21)
Eq. (21) shows that when a thermodynamic system is perturbed with a Gaussian stochastic energy supply thorough its boundary, the internal randomisation of the system (i.e., its entropy fluctuations) is not Gaussian, but is described by a gamma probability density function. This conclusion, which is not evident a priori, demonstrates that the system randomisation due to external perturbations does not necessarily have the same characteristics of the perturbation itself. More in general, this work shows that the approach used in mesoscopic thermodynamics can be extended to thermodynamic systems which do not strictly belong to the category of mesoscopic systems whose internal perturbations are not negligible, but where perturbations are introduced from outside.
[1] R.S. Ingarden, Towards mesoscopic thermodynamics: small systems in higherorder states, Open Syst. Inf. Dyn. 1 (1992) 75. [2] J.M.G. Vilar, J.M. Rubi, Proc. Natl. Acad. Sci. USA 98 (2001) 11081. [3] D. Reguera, J.M. Rubi, J.M.G. Vilar, J. Phys. Chem. B 109 (2005) 21502. [4] J.M. Rubi, Sci. Am. 299 (2008) 62. [5] A. Perez-Madrid, J.M. Rubi, P. Mazur, Physica A 212 (1994) 231. [6] I. Santamaria-Holek, J.M. Rubi, A. Perez-Madrid, New J. Phys. 7 (2005) 35. [7] J.M. Rubi, Mesoscopic thermodynamics, Phys. Scr. T 151 (2012) 014027. [8] I. Pagonabarraga, J.M. Rubi, Physica A 188 (1992) 553. [9] I. Pagonabarraga, A. Perez-Madrid, J.M. Rubi, Physica A 237 (1997) 205. [10] D. Reguera, J.M. Rubi, J. Chem. Phys. 109 (1998) 5987. [11] S. Kjelstrup, J.M. Rubi, D. Bedeaux, J. Theor. Biol. 234 (2005) 7. [12] S. Kjelstrup, J.M. Rubi, D. Bedeaux, Phys. Chem. Chem. Phys. 23 (2005) 4009. [13] D. Bedeaux, S. Kjelstrup, J.M. Rubi, J. Chem. Phys. 119 (2003) 9163. [14] C. Kittel, H. Kroemer, Thermal Physics, 2nd ed., W.H. Freeman and Company, United States of America, 1980.
(15)
where the integration constant C is determined by the normalisation condition. Substituting Eqs. (13) and (14), Eq. (15) becomes:
1
(14)
0
C
γ
where C = C k B ; the integration constant is given by:
References
dη
(18)
(13)
The coefficients A and B are not explicitly dependent on time. The Fokker–Planck equation has a steady-state solution given by:
C
P (β) = C β γ −1 exp(−γ k B T 0 β)
P ( z) = (10)
cV
where γ = c V /k B . The probability density function of temperature can be expressed in terms of the normalised inverse temperature β = 1/k B T , and recalling that dβ = −(1/k B T 2 )dT yields:
and with the further variable change z = γ k B T 0 β yields a gamma distribution:
(9)
T (t )
(17)
(8)
and k B is Boltzmann’s constant. Under these assumptions, the subsystem temperature is described by the following Langevin equation:
dT (t )
(7)
where ξ(t ) is a Gaussian stochastic process with
T0 exp −γ 1 + γ T T
P (β) =
We define the stochastic component of the heat rate as:
Q˙ S (t ) = T (t )ξ(t )
C
(16)