Nonequilibrium Thermodynamics of open driven systems Hao Ge 1 - - PowerPoint PPT Presentation

Nonequilibrium Thermodynamics of

• pen driven systems

Hao Ge

1Biodynamic Optical Imaging Center (BIOPIC) 2Beijing International Center for Mathematical

Research (BICMR) Peking University, China

Laws of thermodynamics

Microscopic reversibility Detailed balance At equilibrium Zeroth law: The definition of temperature First law: Energy conservation Second law: the arrow of time Third law: absolute zero temperature

Q W dU   T Q dS  

Clausius inequality

Evolution of entropy

e i system

dS dS dS     

medium system tot

dS dS dS        



i i i tot i

X J dS T dS T epr T

Detailed balance

  

i i

X J

• I. Prigogine: Introduction to thermodynamics of irreversible processes. 3rd ed. (1967)

T.L. Hill: Free energy transduction in biology. (1977) dSi, dSe and dStot, rather than Si, Se and Stot are the state functions of the internal system. Generalized flux Generalized force Two different perspectives System Medium

T

Two major questions

medium e system

dS epr dS epr dS    

• 1. In steady state, what does the state function

T·dSmedium mean? Total heat dissipation? Can it be used to perform work? It requires a “real driven” perspective and a minimum work argument.

• 2. In the relaxation process towards steady state,

how to distinguish the two origin of nonequilibrium, i.e. nonstationary and non- detailed-balance (driven) of the steady state?

hk d

Q f epr T   

A single biochemical reaction cycle

Spontaneous ATP hydrolysis and related ATP regenerating system.

Equilibrium condition:

2 1 2 1

] [ ] [ ] [ k k k k P ADP ATP

eq i eq eq  



ATP B 

ADP C 

1

k

1 

k

i

P B 

C

2

k

2 

k

(1) (2)

Open driven system: regenerating system keeping the concentrations of ATP , ADP and Pi

1 ] [ ] [ ] [

2 1 2 1

 

  ss i ss ss

P ADP k k ATP k k 

A single biochemical reaction cycle

Heat dissipation

   

.

medium e Pi ADP ATP B Pi C ADP C ATP B d

S T S T h h h h h h h h h h h                 ) 1 (

) 2 (

After an internal clockwise cycle, the traditional heat dissipation during ATP hydrolysis

Could not be calculated only from the dynamics of the internal system.

There is an external step for the regenerating system converting ADP+ Pi back to ATP after each completion of a cycle. The minimum work (non-PV) it has to do is just the free energy difference between ADP+ Pi and ATP , i.e.

Pi ADP ATP

W      

min

The total heat dissipation of such a reaction cycle is . S T S T T k W h h

medium e B ext d d

          log

min

Heat dissipation

) (

min Pi ADP ATP ext d

h h h W h    

The extra heat dissipation

Driven energy of the internal system

Master equation model

No matter starting from any initial distribution, it will finally approach its stationary distribution satisfying

 

1

 



 N j ij ss i ji ss j

k c k c



 

j ij i ji j i

k c k c dt t dc ) ( ) (

Consider a motor protein with N different conformations R1,R2,…,RN. kij is the first-order or pseudo-first-order rate constants for the reaction Ri→Rj. Self-assembly or self-organization

ij eq i ji eq j

k c k c 

Detailed balance

Coupled with energy source

] [ ~ ], [ ~

21 21 12 12

ADP k k ATP k k  

Assume only one of the transition is involved in the energy source, i.e. ATP and ADP . If there is no external mechanism to keep the concentrations of ATP and ADP , then

. ~ ~

2 21 1 12

c c k c c k dt dc dt dc

D T D T

    

Thermodynamic constrains

. ~ ~ log , log ; log

21 12 2 1

k k T k c c T k k k T k

B D T eq T eq D B D T ji ij B j i

               

eq i B i

c T k log   

Boltzmann’s law

) ( ) ( ), ( ) (

eq D D eq T T eq j j eq i i

c c c c      

Heat dissipation

} { ; ... ... log

2 1 min

1 1 2 1 1

i i i i i c k k k k k k T k Q

n i i i i i i i i i i i i B c

n n n n

     





  

  

D T B j i j i ji j ij i B

• pen

d

k t c k t c T k h h k t c k t c T k t h        



 21 2 12 1

) ( ) ( ) ( ) ( ) ( ~

 

. log ~

medium e j i ji ij ji ss j ij ss i B ness d

dS T dS T k k k c k c T k h      





In an NESS, its kinetics and thermodynamics can be decomposed into different cycles (Kirchhoff’s law, Beijing school). The minimum amount of total heat dissipation for each internal cycle

Energy transduction efficiency

A mechanical system coupled fully reversibly to a chemical reactions, with a constant force resisting the mechanical movement driven by the chemical gradient.

mech ness p mech ness d m c

P Te P h J W     



~

min

Transduction from chemical energy to mechanical energy Transduction from mechanical energy to chemical energy

, , ,

min

   

 mech ness p m c

P e J W , , ,

min

   

 mech ness p m c

P e J W 1

min

    

 mech ness p mech m c mech

P Te P J W P  1

min min min

         

   m c ness p m c mech m c

J W Te J W P J W 

The steady-state entropy production is always the total dissipation, which is nonnegative

The evolution of entropy

• pen
• pen
• pen

TS S T H F     ~ ~  . log ;

i i i B

• pen

i i i

c c k S c s S

 

  

. ; ~ ~ T h e dt dS T h e dt S d

• pen

d

• pen

p

• pen
• pen

d

• pen

p

• pen

   

• pen
• pen

S S S   ~

 

 

i i i i i i

c c h H ,  

   

. log ) ( ) ( ) ( ; log ) ( ) ( ) (

ji j ij i j i ji j ij i B

• pen

p ji ij j i ji j ij i B

• pen

d

k c k c k t c k t c k t e k k k t c k t c T k t h

 

 

   

ness d ness d

h h ~ 

Operationally defined heat if we do not know the temperature dependence of 

Enthalpy-entropy compensation

QSS v.s. NESS

. ; ;       

close d close p

• pen

p close d close p close close d close

f Te Te T h e dt dS f dt dF

This reflects the different perspective of Boltzmann/Gibbs and Prigogine: Gibbs states free energy never increase in a closed, isothermal system; while Prigogine states that the entropy production is non-negative in an open system. They are equivalent.

Closed system

Very slow changing environment

Real driven: Housekeeping heat

 

. log ) ( ) ( ) (





 

j i ji ss j ij ss i ji j ij i B hk

k c k c k t c k t c T k t Q

 

log

j i ji ij B ij

s s T k k T k Q   

Housekeeping heat The steady-state entropy difference

log

i j ss j ss i B ss ij

s s c c k S    

The minimum heat dissipation for each cycle could be distributed to each i→j as

ji ss j ij ss i B ss ij ij

k c k c T k S T Q log   

0. ) (    

ss ij ij hk

S T Q t Q   0 ) (t Qhk

No driven (approaching equilibrium state with detailed balance)

Time-independent systems

;

d

f dt dF  

fd: free energy dissipation rate ;

d hk

h Q dt dE   hd: heat dissipation/work out Qhk: house keeping heat/work in

; T h e dt dS

d p 



ep: entropy production rate

Relative entropy

   

; log ) ( ) ( ) ( log ) ( ) ( ) (

ji ij j i ji j ij i B d ji j ij i j i ji j ij i B p

k k k t c k t c T k t h k c k c k t c k t c k t e

 

 

    ；

 

. log ) ( ) ( ) (





 

j i ji ss j ij ss i ji j ij i B hk

k c k c k t c k t c T k t Q

. , ,     

hk d p hk d

Q f Te Q f

ep characterizes total time irreversibility in a Markov process. When system reaches stationary, fd = 0. When system is closed (i.e., no active energy drive, detailed balaned) Qhk = 0. Boltzmann: fd = T∙ep >0 but Qhk=0; Prigogine (Brussel school, NESS): Qhk=T∙ep > 0 but fd=0. fd ≥ 0 in driven systems is “self‐organization”.

Two origins of irreversibility

Time-dependent systems

T t h t e dt t dS

d p

) ( ) ( ) (  

) ( ) ( ) ( t Q t W dt t dU

ex ext

 

) ( ) ( ) ( t f t W dt t dF

d ext

 



 

j ij i ji j i

t k c t k c dt t dc )) ( ) ( ( ) (

) ( ) ( ) ( ) ( ) ( t Q t h t f t Te t Q

ex d d p hk

   

Dissipative work in Jarzynski equality Entropy in Hatano- Sasa equality. We would like to call it intrinsic entropy, which could be defined at individual level. Dissipative work in Jarzynski equality

Two kinds of Second Law

 

  

p d e

T h dt dS

I n non-detailed balance case, the new one is stronger than the traditional one. I n detailed-balance case, they are equivalent.

 

 

d ext f

W dt dF ,  

hk d p

Q f Te T h T Q dt dS Q dt dU W dt dF

d ex ex ext

       

Summary

 Regenerating system approach would distinguish quasi-

steady-state and nonequilibrium-steady-state, and supply an equilibrium thermodynamic foundation for the expression of heat dissipation in nonequilibrium steady state of subsystems, without the need to know “environment”;

Thermodynamic superstructure would explicitly distinguish

Boltzmann and Prigogine’s thesis, and further clarify the two kinds of the Second Law;

 So far, a comprehensive framework for both equilibrium

and nonequilibrium statistical mechanics is proposed.

Acknowledgement

• Prof. Hong Qian

University of Washington Department of Applied Mathematics

Thanks for your attention!