Post on 17-Dec-2015
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
mrubi@ub.edu