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Introduction to Nonequilibrium Thermodynamics

Introduction to Nonequilibrium Thermodynamics
Introduction to Nonequilibrium Thermodynamics: From Onsager to Micromotors Based on the lecture “Nonequilibrium phenomena in micro and nanosystems” taught at Freie Universität Berlin Jan Korbel Faculty of Nuclear Sciences and Physical Engineering, CTU, Prague 6th Student Colloquium and School on Mathematical Physics,Stará Lesná 25. 8. 2012 1 / 23Outline  History Motivation  Introduction to nonequilibrium thermodynamics  Application: Brownian motors  Recent developments in nonequilibrium TD 2 / 23History Motivation  Theory of nonequilibrium thermodynamics originates from the first half of 20. century  It was mainly developed by Onsager, Rayleigh...  Aim: to extend a formalism of equilibrium processes to dissipative or fast processes  Many processes observed in real system exhibit behavior of irreversible processes  Applications: biophysics, nanosystems,... 3 / 23 General laws  Systemspecific response coefficients: c ;c ; ;::: p v T  Description of macroscopic systems  Small fluctuations can be neglected p E N 1 ' 'p (1) E N N  Equilibrium: state of a system, where we cannot observe any change of measurable quantities  Structure of Thermodynamics: Equilibrium thermodynamics Basic notes 4 / 23 General laws  Systemspecific response coefficients: c ;c ; ;::: p v T  Equilibrium: state of a system, where we cannot observe any change of measurable quantities  Structure of Thermodynamics: Equilibrium thermodynamics Basic notes  Description of macroscopic systems  Small fluctuations can be neglected p E N 1 ' 'p (1) E N N 4 / 23 General laws  Systemspecific response coefficients: c ;c ; ;::: p v T  Structure of Thermodynamics: Equilibrium thermodynamics Basic notes  Description of macroscopic systems  Small fluctuations can be neglected p E N 1 ' 'p (1) E N N  Equilibrium: state of a system, where we cannot observe any change of measurable quantities 4 / 23Equilibrium thermodynamics Basic notes  Description of macroscopic systems  Small fluctuations can be neglected p E N 1 ' 'p (1) E N N  Equilibrium: state of a system, where we cannot observe any change of measurable quantities  Structure of Thermodynamics:  General laws  Systemspecific response coefficients: c ;c ; ;::: p v T 4 / 23Equilibrium thermodynamics Laws of thermodynamics  First law (Claussius 1850, Helmholtz 1847): Energy is conserved. dU =QW (2)  Second law (Carnot 1824, Claussius 1854, Kelvin): Heat cannot be fully transformed into work. Q dS (3) T  Third law: We cannot bring the system into the absolute zero temperature in a finite number of steps. 5 / 23Equilibrium thermodynamics Laws of thermodynamics  First law (Claussius 1850, Helmholtz 1847): Energy is conserved. dU =QW (2)  Second law (Carnot 1824, Claussius 1854, Kelvin): Heat cannot be fully transformed into work. Q dS (3) T  Third law: We cannot bring the system into the absolute zero temperature in a finite number of steps. 5 / 23 For quasistatic reversible process we have   X dU S R dS = + Y dX Y = (4) R i i i T X i i Q  From the second law we know that S = R T Q  For irreversible process we get an extra entropy S = + S i T where S 0 i  Entropy production rate: X dS S X i = (5) dt X t i i  aim of Nonequilibrium TD: to compute entropy production rate Nonequilibrium thermodynamics 6 / 23Q  For irreversible process we get an extra entropy S = + S i T where S 0 i  Entropy production rate: X dS S X i = (5) dt X t i i  aim of Nonequilibrium TD: to compute entropy production rate Nonequilibrium thermodynamics  For quasistatic reversible process we have   X dU S R dS = + Y dX Y = (4) R i i i T X i i Q  From the second law we know that S = R T 6 / 23 Entropy production rate: X dS S X i = (5) dt X t i i  aim of Nonequilibrium TD: to compute entropy production rate Nonequilibrium thermodynamics  For quasistatic reversible process we have   X dU S R dS = + Y dX Y = (4) R i i i T X i i Q  From the second law we know that S = R T Q  For irreversible process we get an extra entropy S = + S i T where S 0 i 6 / 23 aim of Nonequilibrium TD: to compute entropy production rate Nonequilibrium thermodynamics  For quasistatic reversible process we have   X dU S R dS = + Y dX Y = (4) R i i i T X i i Q  From the second law we know that S = R T Q  For irreversible process we get an extra entropy S = + S i T where S 0 i  Entropy production rate: X dS S X i = (5) dt X t i i 6 / 23Nonequilibrium thermodynamics  For quasistatic reversible process we have   X dU S R dS = + Y dX Y = (4) R i i i T X i i Q  From the second law we know that S = R T Q  For irreversible process we get an extra entropy S = + S i T where S 0 i  Entropy production rate: X dS S X i = (5) dt X t i i  aim of Nonequilibrium TD: to compute entropy production rate 6 / 23 There exists no unified theory of nonequilibrium thermodynamics.  Near equilibrium exists a linear theory that is universal.  Let us consider a system which we divide into small subsystems. We assume that every system is in local equilibirium a a b b  Total entropy is: S = S (X ) +S (X ) +::: i i  Entropy production rate for a subsystema: a X X dS a a a a a  = = Y X = Y J (6) i i i i dt i i Nonequilibrium thermodynamics Linear thermodynamics 7 / 23 Near equilibrium exists a linear theory that is universal.  Let us consider a system which we divide into small subsystems. We assume that every system is in local equilibirium a a b b  Total entropy is: S = S (X ) +S (X ) +::: i i  Entropy production rate for a subsystema: a X X dS a a a a a  = = Y X = Y J (6) i i i i dt i i Nonequilibrium thermodynamics Linear thermodynamics  There exists no unified theory of nonequilibrium thermodynamics. 7 / 23 Let us consider a system which we divide into small subsystems. We assume that every system is in local equilibirium a a b b  Total entropy is: S = S (X ) +S (X ) +::: i i  Entropy production rate for a subsystema: a X X dS a a a a a  = = Y X = Y J (6) i i i i dt i i Nonequilibrium thermodynamics Linear thermodynamics  There exists no unified theory of nonequilibrium thermodynamics.  Near equilibrium exists a linear theory that is universal. 7 / 23a a b b  Total entropy is: S = S (X ) +S (X ) +::: i i  Entropy production rate for a subsystema: a X X dS a a a a a  = = Y X = Y J (6) i i i i dt i i Nonequilibrium thermodynamics Linear thermodynamics  There exists no unified theory of nonequilibrium thermodynamics.  Near equilibrium exists a linear theory that is universal.  Let us consider a system which we divide into small subsystems. We assume that every system is in local equilibirium 7 / 23Nonequilibrium thermodynamics Linear thermodynamics  There exists no unified theory of nonequilibrium thermodynamics.  Near equilibrium exists a linear theory that is universal.  Let us consider a system which we divide into small subsystems. We assume that every system is in local equilibirium a a b b  Total entropy is: S = S (X ) +S (X ) +::: i i  Entropy production rate for a subsystema: a X X dS a a a a a  = = Y X = Y J (6) i i i i dt i i 7 / 23ab a b  := Y Y is affinity i i i  Affinity deviation from equilibrium TD force  A system brought from equilibrium reacts by creating a current X J = L (7) i ij j j  L nonequilibrium response coefficients ij  Generally are L functions of ’s, but near equilibrium are assumed to ij be constants J ’s are linear functions of ’s i Nonequilibrium thermodynamics Current and Affinity a a  J is generalized current, at equilibrium J = 0 i i 8 / 23 A system brought from equilibrium reacts by creating a current X J = L (7) i ij j j  L nonequilibrium response coefficients ij  Generally are L functions of ’s, but near equilibrium are assumed to ij be constants J ’s are linear functions of ’s i Nonequilibrium thermodynamics Current and Affinity a a  J is generalized current, at equilibrium J = 0 i i ab a b  := Y Y is affinity i i i  Affinity deviation from equilibrium TD force 8 / 23 Generally are L functions of ’s, but near equilibrium are assumed to ij be constants J ’s are linear functions of ’s i Nonequilibrium thermodynamics Current and Affinity a a  J is generalized current, at equilibrium J = 0 i i ab a b  := Y Y is affinity i i i  Affinity deviation from equilibrium TD force  A system brought from equilibrium reacts by creating a current X J = L (7) i ij j j  L nonequilibrium response coefficients ij 8 / 23Nonequilibrium thermodynamics Current and Affinity a a  J is generalized current, at equilibrium J = 0 i i ab a b  := Y Y is affinity i i i  Affinity deviation from equilibrium TD force  A system brought from equilibrium reacts by creating a current X J = L (7) i ij j j  L nonequilibrium response coefficients ij  Generally are L functions of ’s, but near equilibrium are assumed to ij be constants J ’s are linear functions of ’s i 8 / 23 We can rewrite entropy production as X X  = J = L (8) i i ij i j i ij dS  From the second law:  0, which implies detL 0, L  0 ii dt  In case of two currents we get L L L L  0 11 22 21 12 Onsager relations (L. Onsager, Nobel prize 1968): The matrix L is symmetric, i.e. L = L ij ji 2  For two currents: L  L L 11 22 12 S A  It says more than second law of TD: if L = L +L , then   X X X S A S  = L = L +L = L  0: (9) ij i j i j i j ij ij ij ij ij ij Nonequilibrium thermodynamics Onsager relations 9 / 23 In case of two currents we get L L L L  0 11 22 21 12 Onsager relations (L. Onsager, Nobel prize 1968): The matrix L is symmetric, i.e. L = L ij ji 2  For two currents: L  L L 11 22 12 S A  It says more than second law of TD: if L = L +L , then   X X X S A S  = L = L +L = L  0: (9) ij i j i j i j ij ij ij ij ij ij Nonequilibrium thermodynamics Onsager relations  We can rewrite entropy production as X X  = J = L (8) i i ij i j i ij dS  From the second law:  0, which implies detL 0, L  0 ii dt 9 / 23Onsager relations (L. Onsager, Nobel prize 1968): The matrix L is symmetric, i.e. L = L ij ji 2  For two currents: L  L L 11 22 12 S A  It says more than second law of TD: if L = L +L , then   X X X S A S  = L = L +L = L  0: (9) ij i j i j i j ij ij ij ij ij ij Nonequilibrium thermodynamics Onsager relations  We can rewrite entropy production as X X  = J = L (8) i i ij i j i ij dS  From the second law:  0, which implies detL 0, L  0 ii dt  In case of two currents we get L L L L  0 11 22 21 12 9 / 23S A  It says more than second law of TD: if L = L +L , then   X X X S A S  = L = L +L = L  0: (9) ij i j i j i j ij ij ij ij ij ij Nonequilibrium thermodynamics Onsager relations  We can rewrite entropy production as X X  = J = L (8) i i ij i j i ij dS  From the second law:  0, which implies detL 0, L  0 ii dt  In case of two currents we get L L L L  0 11 22 21 12 Onsager relations (L. Onsager, Nobel prize 1968): The matrix L is symmetric, i.e. L = L ij ji 2  For two currents: L  L L 11 22 12 9 / 23Nonequilibrium thermodynamics Onsager relations  We can rewrite entropy production as X X  = J = L (8) i i ij i j i ij dS  From the second law:  0, which implies detL 0, L  0 ii dt  In case of two currents we get L L L L  0 11 22 21 12 Onsager relations (L. Onsager, Nobel prize 1968): The matrix L is symmetric, i.e. L = L ij ji 2  For two currents: L  L L 11 22 12 S A  It says more than second law of TD: if L = L +L , then   X X X S A S  = L = L +L = L  0: (9) ij i j i j i j ij ij ij ij ij ij 9 / 233  Volume scales as L inertial forces, weight,... 2  Surface scales as L friction, heat transfer,... friction 1   for small systems become friction forces important inertia L  For microsystems is the thermalization time very small instant thermalization  In nonequilibrium TD fluctuations cannot be neglected  Laws are the same, but importance of quantities is different  Macromotor: based on inertia and temperature difference  Micromotor: based on random fluctuations Application: Brownian motors Microsystems 10 / 233  Volume scales as L inertial forces, weight,... 2  Surface scales as L friction, heat transfer,... friction 1   for small systems become friction forces important inertia L  For microsystems is the thermalization time very small instant thermalization  Macromotor: based on inertia and temperature difference  Micromotor: based on random fluctuations Application: Brownian motors Microsystems  In nonequilibrium TD fluctuations cannot be neglected  Laws are the same, but importance of quantities is different 10 / 23friction 1   for small systems become friction forces important inertia L  For microsystems is the thermalization time very small instant thermalization  Macromotor: based on inertia and temperature difference  Micromotor: based on random fluctuations Application: Brownian motors Microsystems  In nonequilibrium TD fluctuations cannot be neglected  Laws are the same, but importance of quantities is different 3  Volume scales as L inertial forces, weight,... 2  Surface scales as L friction, heat transfer,... 10 / 23Application: Brownian motors Microsystems  In nonequilibrium TD fluctuations cannot be neglected  Laws are the same, but importance of quantities is different 3  Volume scales as L inertial forces, weight,... 2  Surface scales as L friction, heat transfer,... friction 1   for small systems become friction forces important inertia L  For microsystems is the thermalization time very small instant thermalization  Macromotor: based on inertia and temperature difference  Micromotor: based on random fluctuations 10 / 23Brownian motors Transport in living cells  In living cells we can observe a few types of transport mechanisms  One is transport of kinesin protein with cargo on the actin filament  We can see a directed “walking” of kinesin on the filament  The mechanism is based on nonequilibrium fluctuations Brownian motors 11 / 23 In equilibrium: No. (fluctuations are neglected)  Out of equilibrium: Yes  In order to get some useful work we use spatial and temporal asymmetry (ratchet effect)  Flashing (onoff) ratchet  Rocking ratchet  Correlation ratchet (based on the disruption of fluctuationdissipation theorem)  Chemical ratchet Brownian motors Ratchets Question: Can exist an engine that exploits random fluctuations in order to produce some work 12 / 23 Out of equilibrium: Yes  In order to get some useful work we use spatial and temporal asymmetry (ratchet effect)  Flashing (onoff) ratchet  Rocking ratchet  Correlation ratchet (based on the disruption of fluctuationdissipation theorem)  Chemical ratchet Brownian motors Ratchets Question: Can exist an engine that exploits random fluctuations in order to produce some work  In equilibrium: No. (fluctuations are neglected) 12 / 23 In order to get some useful work we use spatial and temporal asymmetry (ratchet effect)  Flashing (onoff) ratchet  Rocking ratchet  Correlation ratchet (based on the disruption of fluctuationdissipation theorem)  Chemical ratchet Brownian motors Ratchets Question: Can exist an engine that exploits random fluctuations in order to produce some work  In equilibrium: No. (fluctuations are neglected)  Out of equilibrium: Yes 12 / 23 Flashing (onoff) ratchet  Rocking ratchet  Correlation ratchet (based on the disruption of fluctuationdissipation theorem)  Chemical ratchet Brownian motors Ratchets Question: Can exist an engine that exploits random fluctuations in order to produce some work  In equilibrium: No. (fluctuations are neglected)  Out of equilibrium: Yes  In order to get some useful work we use spatial and temporal asymmetry (ratchet effect) 12 / 23Brownian motors Ratchets Question: Can exist an engine that exploits random fluctuations in order to produce some work  In equilibrium: No. (fluctuations are neglected)  Out of equilibrium: Yes  In order to get some useful work we use spatial and temporal asymmetry (ratchet effect) Types of ratchets  Flashing (onoff) ratchet  Rocking ratchet  Correlation ratchet (based on the disruption of fluctuationdissipation theorem)  Chemical ratchet 12 / 23 The transport is based on switching on and off of an periodic, asymmetric potential  Examples of potentials: asymmetric sawtooth,  1  V (x) = sin(x) + sin 2x + 4 4   2 x  When the potential is off diffusion: p(x;t)' exp 2Dt  When the potential is on particles tend to get to minimums localization: p(x)' exp( V (x))  Because the potential is periodic, no force is present on average  We can observe a particle flow Brownian motors Flashing ratchet 13 / 23  2 x  When the potential is off diffusion: p(x;t)' exp 2Dt  When the potential is on particles tend to get to minimums localization: p(x)' exp( V (x))  Because the potential is periodic, no force is present on average  We can observe a particle flow Brownian motors Flashing ratchet  The transport is based on switching on and off of an periodic, asymmetric potential  Examples of potentials: asymmetric sawtooth,  1  V (x) = sin(x) + sin 2x + 4 4 13 / 23 Because the potential is periodic, no force is present on average  We can observe a particle flow Brownian motors Flashing ratchet  The transport is based on switching on and off of an periodic, asymmetric potential  Examples of potentials: asymmetric sawtooth,  1  V (x) = sin(x) + sin 2x + 4 4   2 x  When the potential is off diffusion: p(x;t)' exp 2Dt  When the potential is on particles tend to get to minimums localization: p(x)' exp( V (x)) 13 / 23Brownian motors Flashing ratchet  The transport is based on switching on and off of an periodic, asymmetric potential  Examples of potentials: asymmetric sawtooth,  1  V (x) = sin(x) + sin 2x + 4 4   2 x  When the potential is off diffusion: p(x;t)' exp 2Dt  When the potential is on particles tend to get to minimums localization: p(x)' exp( V (x))  Because the potential is periodic, no force is present on average  We can observe a particle flow 13 / 23Brownian motors Flashing ratchet 14 / 23 We use again the asymmetric potential, but instead of switching on and off, we tilt the potential a little bit: V (x;t) = V (x) +c x sin(c t) (10) 0 1 2  Again, due to asymmetry of the potential is the current produced with zero average force. Brownian motors Rocking ratchet 15 / 23Brownian motors Rocking ratchet  We use again the asymmetric potential, but instead of switching on and off, we tilt the potential a little bit: V (x;t) = V (x) +c x sin(c t) (10) 0 1 2  Again, due to asymmetry of the potential is the current produced with zero average force. 15 / 23 Motivated by biological background, another possibility how to force the particle to diffuse, is to give it some additional energy, so it can get from the minimum of the potential,  For that we use a chemical reaction ATP ADP +P (11)  Chemical ratchet is kind of flashing ratchet, where the energy to switch of the potential is from the reaction of ATP Brownian motors Chemical ratchet 16 / 23 For that we use a chemical reaction ATP ADP +P (11)  Chemical ratchet is kind of flashing ratchet, where the energy to switch of the potential is from the reaction of ATP Brownian motors Chemical ratchet  Motivated by biological background, another possibility how to force the particle to diffuse, is to give it some additional energy, so it can get from the minimum of the potential, 16 / 23 Chemical ratchet is kind of flashing ratchet, where the energy to switch of the potential is from the reaction of ATP Brownian motors Chemical ratchet  Motivated by biological background, another possibility how to force the particle to diffuse, is to give it some additional energy, so it can get from the minimum of the potential,  For that we use a chemical reaction ATP ADP +P (11) 16 / 23Brownian motors Chemical ratchet  Motivated by biological background, another possibility how to force the particle to diffuse, is to give it some additional energy, so it can get from the minimum of the potential,  For that we use a chemical reaction ATP ADP +P (11)  Chemical ratchet is kind of flashing ratchet, where the energy to switch of the potential is from the reaction of ATP 16 / 23 Efficiency is defined as a ratio between the performed work and consumed energy W W  = = (12) Q Q  We define the chemical force, which is nothing else than difference between chemical potentials,  =  . The L R consumed energy per unit time is Q = r, where r is chemical reaction rate.  Similarly we obtain the performed work per unit time, which is W = f v, where f is a external force and v is the velocity of ext ext particles. The efficiency is then f v ext  = (13) r Brownian motors Efficiency of a Chemical ratchet 17 / 23 We define the chemical force, which is nothing else than difference between chemical potentials,  =  . The L R consumed energy per unit time is Q = r, where r is chemical reaction rate.  Similarly we obtain the performed work per unit time, which is W = f v, where f is a external force and v is the velocity of ext ext particles. The efficiency is then f v ext  = (13) r Brownian motors Efficiency of a Chemical ratchet  Efficiency is defined as a ratio between the performed work and consumed energy W W  = = (12) Q Q 17 / 23 Similarly we obtain the performed work per unit time, which is W = f v, where f is a external force and v is the velocity of ext ext particles. The efficiency is then f v ext  = (13) r Brownian motors Efficiency of a Chemical ratchet  Efficiency is defined as a ratio between the performed work and consumed energy W W  = = (12) Q Q  We define the chemical force, which is nothing else than difference between chemical potentials,  =  . The L R consumed energy per unit time is Q = r, where r is chemical reaction rate. 17 / 23Brownian motors Efficiency of a Chemical ratchet  Efficiency is defined as a ratio between the performed work and consumed energy W W  = = (12) Q Q  We define the chemical force, which is nothing else than difference between chemical potentials,  =  . The L R consumed energy per unit time is Q = r, where r is chemical reaction rate.  Similarly we obtain the performed work per unit time, which is W = f v, where f is a external force and v is the velocity of ext ext particles. The efficiency is then f v ext  = (13) r 17 / 23 The efficiency for linear regime has the form 2 L a +L a 11 12  = (14) L a +L 21 22 where a = f =. ext   The maximal efficiency is given by the relation = 0 and the a 2 L 12 maximal value is in terms of  = : L L 11 22 p 2 1 1   = (15) max  Brownian motors Efficiency of a Chemical ratchet  Near to equilibrium we can consider a linear thermodynamics, which means that currents are linear functions of forces v = L f +L  11 ext 12 r = L f +L  21 ext 22 18 / 23  The maximal efficiency is given by the relation = 0 and the a 2 L 12 maximal value is in terms of  = : L L 11 22 p 2 1 1   = (15) max  Brownian motors Efficiency of a Chemical ratchet  Near to equilibrium we can consider a linear thermodynamics, which means that currents are linear functions of forces v = L f +L  11 ext 12 r = L f +L  21 ext 22  The efficiency for linear regime has the form 2 L a +L a 11 12  = (14) L a +L 21 22 where a = f =. ext 18 / 23Brownian motors Efficiency of a Chemical ratchet  Near to equilibrium we can consider a linear thermodynamics, which means that currents are linear functions of forces v = L f +L  11 ext 12 r = L f +L  21 ext 22  The efficiency for linear regime has the form 2 L a +L a 11 12  = (14) L a +L 21 22 where a = f =. ext   The maximal efficiency is given by the relation = 0 and the a 2 L 12 maximal value is in terms of  = : L L 11 22 p 2 1 1   = (15) max  18 / 23Brownian motors Efficiency of a Chemical ratchet 2  The maximal efficiency we get for L = L L which means 11 22 12 maximal permissible coupling of currents from second law of thermodynamics, the efficiency is therefore = 1  In comparison to macromotors, where the efficiency is limited T c by 1 , here is no restriction to maximal efficiency and T h micromotors have usually much higher efficiency than macromotors. 19 / 23 The second law of TD tells us, that entropy production is always nonnegative  The second law is nevertheless a statistical statement which holds only in thermodynamical limit  For small systems driven out of equilibrium we can expect some entropy fluctuations that can be may also negative  The quantification gives us Fluctuation theorem (Evans, Cohen, Morris, 1993)  P( = A) t = exp(At) (16)  P( =A) t  where  is timeaveraged irreversible entropy production. t Recent developments of nonequilibrium TD Fluctuation theorem 20 / 23 For small systems driven out of equilibrium we can expect some entropy fluctuations that can be may also negative  The quantification gives us Fluctuation theorem (Evans, Cohen, Morris, 1993)  P( = A) t = exp(At) (16)  P( =A) t  where  is timeaveraged irreversible entropy production. t Recent developments of nonequilibrium TD Fluctuation theorem  The second law of TD tells us, that entropy production is always nonnegative  The second law is nevertheless a statistical statement which holds only in thermodynamical limit 20 / 23 The quantification gives us Fluctuation theorem (Evans, Cohen, Morris, 1993)  P( = A) t = exp(At) (16)  P( =A) t  where  is timeaveraged irreversible entropy production. t Recent developments of nonequilibrium TD Fluctuation theorem  The second law of TD tells us, that entropy production is always nonnegative  The second law is nevertheless a statistical statement which holds only in thermodynamical limit  For small systems driven out of equilibrium we can expect some entropy fluctuations that can be may also negative 20 / 23Recent developments of nonequilibrium TD Fluctuation theorem  The second law of TD tells us, that entropy production is always nonnegative  The second law is nevertheless a statistical statement which holds only in thermodynamical limit  For small systems driven out of equilibrium we can expect some entropy fluctuations that can be may also negative  The quantification gives us Fluctuation theorem (Evans, Cohen, Morris, 1993)  P( = A) t = exp(At) (16)  P( =A) t  where  is timeaveraged irreversible entropy production. t 20 / 23 The importance of the theorem is in the fact that FT is valid for all systems arbitrarly far from equilibrium  A corollary of FT is Second law inequality that says    0 8t; (17) t so ensemble average of entropy production is always positive Recent developments of nonequilibrium TD Fluctuation theorem  With an increasing time or size of the system, negative fluctuations are exponentially supresed. But for small scales and time intervals can negative fluctuations be observed (and already have been measured) 21 / 23 A corollary of FT is Second law inequality that says    0 8t; (17) t so ensemble average of entropy production is always positive Recent developments of nonequilibrium TD Fluctuation theorem  With an increasing time or size of the system, negative fluctuations are exponentially supresed. But for small scales and time intervals can negative fluctuations be observed (and already have been measured)  The importance of the theorem is in the fact that FT is valid for all systems arbitrarly far from equilibrium 21 / 23Recent developments of nonequilibrium TD Fluctuation theorem  With an increasing time or size of the system, negative fluctuations are exponentially supresed. But for small scales and time intervals can negative fluctuations be observed (and already have been measured)  The importance of the theorem is in the fact that FT is valid for all systems arbitrarly far from equilibrium  A corollary of FT is Second law inequality that says    0 8t; (17) t so ensemble average of entropy production is always positive 21 / 23 In thermodynamics can be for quasistatic process derived an inequality between free energy and work F W (18)  It is possible to derive a generalization of this inequality for arbitrary processes (not only “slow”) from the fluctuation theorem  the relation is called Jarzynski equality (Jarzynski, 1997)     F W exp = exp (19) k t k t B B  The line indicated all possible realizations of an external process that takes the system from equilibrium state A to equilibrium state B. States in between these points do not have to be equilibrium states. Recent developments of nonequilibrium TD Jarzynski equality 22 / 23 It is possible to derive a generalization of this inequality for arbitrary processes (not only “slow”) from the fluctuation theorem  the relation is called Jarzynski equality (Jarzynski, 1997)     F W exp = exp (19) k t k t B B  The line indicated all possible realizations of an external process that takes the system from equilibrium state A to equilibrium state B. States in between these points do not have to be equilibrium states. Recent developments of nonequilibrium TD Jarzynski equality  In thermodynamics can be for quasistatic process derived an inequality between free energy and work F W (18) 22 / 23 the relation is called Jarzynski equality (Jarzynski, 1997)     F W exp = exp (19) k t k t B B  The line indicated all possible realizations of an external process that takes the system from equilibrium state A to equilibrium state B. States in between these points do not have to be equilibrium states. Recent developments of nonequilibrium TD Jarzynski equality  In thermodynamics can be for quasistatic process derived an inequality between free energy and work F W (18)  It is possible to derive a generalization of this inequality for arbitrary processes (not only “slow”) from the fluctuation theorem 22 / 23 The line indicated all possible realizations of an external process that takes the system from equilibrium state A to equilibrium state B. States in between these points do not have to be equilibrium states. Recent developments of nonequilibrium TD Jarzynski equality  In thermodynamics can be for quasistatic process derived an inequality between free energy and work F W (18)  It is possible to derive a generalization of this inequality for arbitrary processes (not only “slow”) from the fluctuation theorem  the relation is called Jarzynski equality (Jarzynski, 1997)     F W exp = exp (19) k t k t B B 22 / 23Recent developments of nonequilibrium TD Jarzynski equality  In thermodynamics can be for quasistatic process derived an inequality between free energy and work F W (18)  It is possible to derive a generalization of this inequality for arbitrary processes (not only “slow”) from the fluctuation theorem  the relation is called Jarzynski equality (Jarzynski, 1997)     F W exp = exp (19) k t k t B B  The line indicated all possible realizations of an external process that takes the system from equilibrium state A to equilibrium state B. States in between these points do not have to be equilibrium states. 22 / 23Robert Zwanzig. Nonequilibrium Statistical Mechanics. Oxford University Press, USA, March 2001. P. Hänggi, F. Marchesoni, and F. Nori. Brownian motors. Annalen der Physik, 14(13):51–70, 2005. Andrea Parmeggiani, Frank Jülicher, Armand Ajdari, and Jacques Prost. Energy transduction of isothermal ratchets: Generic aspects and specific examples close to and far from equilibrium. Phys. Rev. E, 60:2127–2140, Aug 1999. Denis J. Evans, E. G. D. Cohen, and G. P. Morriss. Probability of second law violations in shearing steady states. Phys. Rev. Lett., 71:2401–2404, Oct 1993. Harvard Biovisions. Molecular machinery of life: Online video. http://www.youtube.com/watchv=FJ4N0iSeR8U, February 2011. Thank you for attention 23 / 23
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