the mesoscopic dynamics of thermodynamic systems j.m. rubi
TRANSCRIPT
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