Slow Energy Relaxation in Complex Systems

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Slow Energy Relaxation in Complex Systems

• Francesco Piazza
• Laboratore de Biophysique Statistique
• EPF-Lausanne, Switzerland

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Energy relaxation in a 1d chain of non-linear
oscillators
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(No Transcript)
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Slow relaxation phenomena some examples

• Well-known slow kinetics in glasses and
  super-cooled liquids

• Protein folding

• Slow relaxation of local energy fluctuations in
  proteins
• (e.g. after photo-dissociation of CO group in
  Myoglobin).
• (Sabelko et al. PNAS 1999)

• Slow relaxation of global heat transfer to the
  environment in metallic nano-clusters.
• (Hu Hartland, J.Phys.Chem. B 2002)

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The formal explanation
There exists a spectrum of relaxation rates
associated with the degrees of freedom of the
system. The latter relax exponentially
If the response kernel g(x) is a sharply peaked
function, themeasured relaxation will be an
exponential decay dominatedby the d.o.f. in the
vicinity of lt x gt . Otherwise, the whole pool of
degrees of freedom will contribute and
slower-than-exponential behaviours may
arise(power law, stretched exponential).
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Possible physical scenarios
It merely rephrases the problem. Unlikely under
general circumstances.
It applies to the dynamics offar-from-equilibrium
perturbations only.
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A critical view
The following chain of implications is commonly
maintained
But how complex should be a complex system?
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The simplest complex system
A linear chain of beads and springs is
thermalized at T T1. Then it is put in contact
at its edges with thermal baths at T T2 lt T1.
Whatis the energy decay law like?
First solve the problem in the simplest case T2
0.
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Analytical solution
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To first order in ?, we obtain
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The important message
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Relaxation from equilibrium
f fraction of particles in contact with
the environment The cross-over exists also
in the thermodynamic limit
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Energy relaxation in nano-systems
Nano-systems live immersed in a medium and have
large values of surface-to-volume ratio.
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The problem
We want to study the relaxation of the system
from a temperature T1 to a lower temperature
T2 The individual units (residues, atoms) are
assigned a local surface fraction f (bulk and
surface) We describe the stochastic dynamics à
la Langevin. The damping constants are taken to
be proportional to f The amplitudes of the
fluctuating forces are set accordingly, in
accordance to the FD theorem.
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The simplest model of a nano-system a network of
beads and springs
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The harmonic approximation
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The Langevin dynamics
A simple tool to introduce the coupling with the
medium. Particles displacements are governed by
stochastic equations of motion of the Langevin
type
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Equations of motion in matrix form
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The relaxation spectrum
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Relaxation dynamics in biological systems
This broad topic encompasses some of the
fundamental processes of molecular biology, such
as the dynamics of relaxation and redistribution
of energy released at specific sites in a
protein structure after, e.g.

• absorption of electromagnetic radiation
  (conformational changes induced
• in rhodopsin after absorption of a visible
  photon),

• completion of an exothermic chemical reaction
  (hydrolysis of an ATP
• molecule into ADP, the basic fuelling
  mechanism for functioning of molecular
• motors).

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Relaxation in a metal nano-cluster

• Relaxation after excitation with laser light has
  twocharacteristic time scales
• fast (lt ps) dynamics of e-e equilibration
• slow (gt ps) dynamics of heat dissipation to the
• environment

• Heat dissipation from bio-functionalized
  particles used to selectively kill cells or to
  study protein denaturation

• Heat dissipation is also an important issue in
  laser-induced annealing and size and shape
  transformation of metal particles.

Experimental evidence for slow (stretched
exponential) relaxation(M. Hu and G. V.
Hartland, J. Phys. Chem. B. 106, 7029 (2002))
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The Fokker Planck formalism
Such phenomena of relaxation dynamics and related
ones can be studied analytically by solving the
Fokker-Planck equation associated with the
Langevin elastic network model of the system.
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Fokker-Planck formulation of the problem
is the probability that the system is described
by theset of displacements and velocities Y at
time t if itsinitial configuration at time t 0
was Y(0)
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The solution
where G is the propagator matrix and
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The evolution law for the correlation matrix

• C(0) describes the initial excitation.

the relaxation depends only on the temperature
difference
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The energy decay
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Myoglobin
Sample nano-cluster
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Conclusions

• A simple extended harmonc system naturally
  develops a spectrum
• of relaxation rates when interacting with
  the environment through its surface

• The response to an energy excitation is a
  superposition of many exponentially decaying
  channels the result is a rich, non-exponential
  behaviour.
• a first portion of the accumulated energy decays
  exponentially with a rate constant characteristic
  of the fastest d.o.f.
• Then the decay crosses over to an integrated
  non-exponential regime, which is the full
  super-position of all relaxing d.o.f. The
  cross-over should be observable in nano-systems
  and is also predicted to be observable in the
  thermodynamic limit.

• The integrated decay is a power law of the
  type E?d/2 in dimension d for ordered
  structures with nearest-neighbour interactions.
  It is well approximated by a stretched
  exponential law in more complex or disordered
  structures.

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Co-workers

• Roberto Livi, Università di Firenze, IT
• Stefano Lepri, ISC-CNR, IT
• Paolo De Los Rios, EPFL, Lausanne, CH
• Yves-Henri Sanejouand, ENS, Lyon, FR

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