The mesoscopic dynamics of thermodynamic systems J.M. Rubi.

The mesoscopic dynamics of thermodynamic systems

J.M. Rubi

Cluster

Polymer

Single molecule

Pump

Biological cells

Protein

Atomic Mesoscopic

Is thermodynamics applicable to nanosystems?

Peculiar features:

1.Thermodynamic limit not fulfilled. Free energy contains more

contributions2

3( , ) ( , )G N T P N h T P

Surface contribution;N N     G  

2. Fluctuations can be larger than average values

A A A

Macroscopic: continuum

A A 1A

A

thermodynamic value

fluctuation

Diffusion

J D

Fick

i) Large scalesii) Long times

Description in terms of average values

Jt

Thermodynamics of diffusion

Tds d

1J

T x

; /L

J D L TT x

Dt x x

Gibbs; local equilibrium

x:center of mass

:size, others

Local equilibrium:

( ) ( ) ( ) ( )Tds x x d x Fd x

Force

Mesoscale local equilibrium:

( ) ( ) ( )Tds x x dP x

( ) ( )x P x d

Single molecule

Mesoscopic thermodynamics

( , ) ( , ) ( , )Tds x v x v dP x v21

ln ( , )2

kTP x v v

m

Assumption: the system undergoes a diffusionprocess in (x,v)-space

Gibbs equation:

Local equilibrium in (x,v)-space

lnS k P Pd

Probability conservation:

x vJ JP

t x v

Entropy production:

0x vJ Jx v

Currents:x xx xv

v vx vv

J L Lx v

J L Lx v

Onsager relation:

xv vxL L

Currents

2

x

v

DJ v P

v

D v DJ P

x v

0

/

xx

xv

vv

L

L P

L P

2

P v DvP P

t x v v

Kramers

Regimes

0 ; 0x vJ J     

0, 0x vJ J    

0x vJ J Equilibrium:

Local equilibrium

Gaussian, T

Far from equilibrium

Fick

x

PJ D

x

Nonlinear regime

MNET can provide nonlinear equations for the currents

Two types of nonlinearities:

i) In the transport coefficientsii) In the currents

(Q)

Q

1 2

Q1 Q0 Q2

NET: two-state system

( )Q( )Q

1 2

quasi-equilibrium at each well

Examples: chemical reactions,nucleation, adsorption, active transport, thermoionic emission, etc.

NET description

1JA

T

2 1( )L L

J AT T

Law of mass action

2 1

(1 )A

kT kT kT LJ D e e D e A

T

Conclusion: NET only accounts for the linear regime

linearization

intermediateconfigurations0 1

….

The process is described at short time scales. A local value of the potential corresponds to a configuration at a reaction coordinate

enzyme

ions

Mesoscopic thermodynamics

( ) kT kT kT kTL kLJ e e De e

T P

2 2

1 1( ) kT kTJ t d Je D d e

The activation process is viewed as a diffusion process along a reaction coordinate

From local to global:

2 1

2 1( )kT kTJ D e e D z z

...d

Nucleation kinetics

Basic scenario:

melted crystal

Metastable phase

Order parameter

embryo

:

: ( , , )

Cluster at rest x n

Cluster inabath x n v

Transport throughprotein channels

B

P P D SD P

t x x k x

0 2

1( )

(1 ( ) )D x D

y x

Entropic barrier

Scaling law

Polymer crystallization

embryopattern

0 1D Dp

20

1( , ) ( ) ( )( )

2n u n m n u v

Sheared melt

Translocation of a biomolecule

Conclusions

• MNET offers a unified and systematic scheme to analyze irreversible processes taking place at the nano-scale.

• It can be used in the description of the two basic irreversible processes: transport and activation.

• Applications to: transport in materials and in biology, chemical and biochemical kinetics, adsorption, thermoionic emission, spin flip processes, etc.

References

• A. Perez-Madrid, J.M. Rubi and P. Mazur, Physica A 212, 231 (1994)

• J.M. Vilar and J.M. Rubi, Proc. Natl. Acad. Sci., 98, 11081 (2001)

• D. Reguera, J.M. Rubi and J.M. Vilar, J. Phys. Chem. B, 109, 21502 (2005) Feature Article

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