Transcript Entropy Production in a System of Coupled Nonlinear Driven

Entropy Production in a
System of Coupled Nonlinear
Driven Oscillators
Mladen Martinis, Vesna Mikuta-Martinis
Ruđer Bošković Institute, Theoretical Physics Division
Zagreb, Croatia
MATH/CHEM/COMP, Dubrovnik-2006
Motivation
Nonequilibrium
thermodynamics of complex
biological networks
What are the thermodynamic links between biosphere and
environment?
How to bring nonequilibrium thermodynamics to the same level of
clarity and usefulness as equilibrium thermodynamics?
 Energy balance analysis
 Entropy production as a measure of Bio  Env interaction
"A violent order is disorder; and a great
disorder is an order. These two things
are one.“
Wallace Stevens, Connoisseur of Chaos, 1942
 Non-equilibrium may be a source of order
 Irrevesible processes may lead to disspative structures
 Order is a result of far-from-equilibrium (dissipative)
systems trying to maximise stress reduction.
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Equation of balance
New
=
Old
+
(Δe = in – out)
Δ
=
In
-
(Δi = Gen – Con)
Out
Δ = Δe + Δi
Δ
+
Gen
-
Con
= balance
Gen = Generation
Con = Consumption
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Environment
Ein
Energy balance
System
Δ E = Eout - Ein
Eout
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1st law of thermodynamics
(Energy balance equation)
Closed system (no mass transfer)
ΔE = Eout – Ein = ΔQ – ΔW
E = U + Ek + Ep
Open system (mass transfer included)
ΔE = Eout – EEin = ΔQ – ΔWs
out
E = H + Ek + Ep
U  internal energy,
H = U + PV  enthalpy
Ek  kinetic energy
Ep  potential energy
ΔQ  heat flow
ΔW  work
ΔWs  work to make things flow
Energy
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Entropy balance
Environment
Sin
ΔeS = Sout - Sin
System
ΔS = ΔiS + ΔeS
ΔiS ≥ 0
ΔiS  entropy production(EP)
Sout
EP ?
MaxEP
MinEP
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2nd law of thermodynamics
• Entropy production (diS/dt)
• dS = deS + diS with diS ≥ 0
• Entropy production includes many effects:
dissipation, mixing, heat transfer, chemical
reactions,...
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Coupled oscillators
 Many (quasi)periodic phenomena in physics,
chemistry, biology and engineering can be
described by a network of coupled oscillators.
 The dynamics of the individual oscillator in the
network, can be either regular or complicated.
 The collective behavior of all the oscillators in
the network can be extremely rich, ranging from
steady state (periodic oscillations) to chaotic or
turbulent motions.
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Biological oscillators

It is well known that cells, tissues and
organs behave as nonlinear oscillators.

By the evolution of the organism,they are
multiple hierarchicaly and functionally
interconnected
→ complex biological network.
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Biological network
Graph
1
11
3
4
Graph theory
2
2
4
5
g 34
5
6
3
S
6
Graph with weighted edges → network
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Network theory
7
G = {gij} connectivity matrix
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Biological homeostasis
(Dynamic self-regulation)
 Homeostasis (resistance to change) is the property of
an open system, (e.g. living organisms), to regulate its
internal physiological environment:
 to maintain its stability under external varying conditions,
 by means of multiple dynamic equilibrium adjustments,
controlled by interrelated negative feedback regulation
mechanisms.
 Most physiological functions are mainteined within
relatively narrow limits
( → state of physiological homeostasis).
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What is feedback ?
It is a connection between the output of a
system and its input
(  effect is fed back to cause ).
Feedback can be
• negative (tending to stabilise the system 
order) or
• positive (leading to instability  chaos).
Feedback results in nonlinearities
leading to unpredictability.
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Negative feedback control
stabilizes the system
(It is a nonlinear process)
message
Receptor
Effector
Corrective
response
Bf increses
No change in Bf
Bio-factor
Bio-factor
Oscillations around equilibrium
Bf decreases
Corrective
response
message
Receptor
Effector
Osmoregulation, Sugar in the blood regulation, Body temperature regulation14
Coupled nonlinear oscillators
Each oscillating unit (cell, tissue, organ, ...)
is modelled as a nonlinear oscillator with
a globally attracting limit cycle (LC).
The oscillators are weakly coupled gij , and
their natural frequencies ωi are randomly
distributed across the population with
some probability density function (pdf).
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Coupled nonlinear oscillators
(Kuramoto model)
• Given natural frquencies ωi
• Given couplings gij
ωi, gij

Phase transition
Synchronization
Self-organization
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Coupled nonlinear oscillators
dxk/dt = Fk(xk, ck, t) + Σ gik (xi, xk)
i
k
xk = the state vector of an oscillator
xk = (x1k, x2k), k = 1,2, ..., N
gik = coupling function
Fk = intradynamics of an oscillator
Diffusion coupling: gik = μ ik (xi – xk); μ ik = NxN matrix
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Complex phase space
Standard phase space:
s(t), sn = biological signal
dts(t), sn+1 = rate of change
dts(t) = f(s,t) or sn+1 = f(sn)
dts
sn+1
Free oscillator
dt2s + ω2s = 0, s(t) = Acos( ωt + φ )
Complex phase space:
s,sn
Im z
z
z(t) = ωs(t) – i dts(t)
dtz(t) = F(z, z*, t) or zn+1 = F(zn, z*n)
Free oscillator: dtz = i ω z , |z|2 = const
z = r e iθ , dtr = 0, dt θ = ω
ωs = Re z = r cosθ
r
z - plane
Re z = r cosθ
Im z = r sinθ
θ
Re z
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Limit Cycle Oscillator(LCO)
negative feedback effect
Example of LCO:
dtz = (a2 + iω - |z|2)z
dtr = (a2 – r2)r, dt θ = ω
ωs(t) = r(t)cos θ(t)
r0>a
z-plane
r0<a
a
Limit cycle
Solution: r(t) =a/u(t), θ(t) = ωt + θ0
u(t) = [1 – (1- u02)exp(-2a2t)]½
ωs(t) = (a/u(t))cos( ωt + θ0), ω = 2π / T
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Limit cycle property
2,5
Limit cycle oscillator
2,0
r0 > 1
r(t)
1,5
r1(t)
r2(t)
1,0
Limit cycle line
0,5
r0 < 1 1
0,0
0
0,5
1
1,5
2
3
4
5
t - time
r(t) = [1 – (1- r0 -2 )exp(-2t)]-½
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Oscillating signal
1,5
Circadian signal
r0 < 1
1
r(t)cosθ(t)
0,5
time in hours
0
0
5
10
15
20
25
30
-0,5
-1
-1,5
ωs(t) = r(t)cosθ(t), ω = π /12
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Limit cycle in the z - plane
1,5
Limit cycle
1
z - plane
0,5
0
-1,5
-1
-0,5
0
0,5
1
1,5
-0,5
-1
-1,5
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Entropy production
in a driven LC oscillator
•
Biological systems are generically out of equilibrium.
•
In an environment with constant temperature the source of nonequilibrium are usually mechanical (external forces) or chemical
(imbalanced reactions) stimuli with stochastic character of the nonequilibrium processes.
Stochastic (Langevin) description of a driven LC oscillator representing
stochastic trajectory (dts(t), s(t))) in (r, θ)-phase space
dtr(t) = (a2 – r2)r + ς(t), dtθ(t) = ω
ς(t)  Gaussian white noise
< ς(t) ς(t’)> = 2dδ(t – t’)
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Non- equilibrium entropy
Si(t) = - ∫rdr p(r,t) lnp(r,t) ≡ <si(t)>
dtSe
si(t) = - lnp(r,t),
dtse(t) = dtq(t)/ T = (a2 – r2)r dtr, D = T
dtSi
p(r,t) is the probability to find the LCO in the state r
dtS = dtSi + dtSe ≥ 0
p(r,t) is the solution of the the Fokker-Planck equation
with a given initial condition p(r,0) = p0(r)
∂tp(r,t) = - ∂rj(r,t) = - ∂r[(a2 – r2)r - D∂r]p(r,t)
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Applications
Biological Rhythms (BRs)



BRs are observed at all levels of living
organisms.
BRs can occur daily, monthly,
or seasonally.
Circadian (daily) rhythms (CRs)
vary in length from species to species
(usually lasts approximately 24 hours).
Biological Clocks (BCs)

Biological clocks are responsible for
maintaining circadian rhythms, which affect
our sleep, performance, mood and more.
Circadian clocks enhance the fitness of an
organism by improving its ability to adapt to
environmental influences, specifically daily
changes in light, temperature and humidity.
Modelling circadian rhythmus
as coupled oscillators
1. Blood pressure circadian
2. Heart rate circadian
3. Body temperature circadian
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Three coupled oscillators
Single oscillator : dt2x3(t) = - ω32 x3 + ...
BT
A
HR
BP
3
3
B
C
1
2
1
g12
 g11 g12 g13 


g   g 21 g 22 g 23 
g g g 
 31 32 33 
External
stimuli
g23
2
g12
3
g31
D
Coupling matrix
gkk = 0
gjk ≠ gkj
g23
1
2
g12
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Coupled Limit Cycle Oscillators
Linear coupling model*
g31
B
T
H
R
*Aronson et al., Physica D41 (1990) 403
g23
BP
g12
Fkext
k = HR, BP, BT
gkj = - gjk , gkj = Kk δkj
Kk ≥ 0 coupling strength
zk(t) = rk(t) e iθk(t)
dtzk(t) = (ak + iωk - |zk(t)|2)zk(t) + Σgkj (zj(t) – zk(t)) - i Fkext(t)
There are six (6) first order differential equations to be solved
for a given initial conditions (rk(0), θk(0); k = 1, 2, 3)
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Consequences
• Coupled limit cycle oscillator model has variety of
stationary and nonstationary solutions which
depend on the coupling K, the limit cycle radius a
and the frequency differencies ∆kj = |ωk – ωj|.
• Weak coupling (K ~ 0): the oscillators behave as
independent units , subjected each to the
influence of the external stimuli (Fext (t)).
• With increasing coupling (K> 1) two important
classes of stationary solutions are possible:
– The amplitude death (r1, r2 or r3 → 0 as t → ∞ )
– The frequency locking (synchronization)
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Conclusion
• We have developed a mathematical models of BP, HR and BT
circadian oscillations using the coupled LC oscillators approach.
• Coupled LC oscillator-model can have variety of stationary and
nonstationary solutions which depend on the coupling K, the limit
cycle radius a and the frequency differencies ∆kj = |ωk – ωj|.
• Weakly coupled oscillators behave as independent units but with
coupled phases.
• They are subjected each to the influence of the external
disturbancies (Fext (t)) which can change circadian organization of
the organism and become an important cause of morbidity.
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END
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Self-organization
Self-organization in biological systems
relies on functional interactions between
populations of structural units (molecules,
cells, tissues, organs, or organisms).
.
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Synchronization
There are several types of synchronization :
•
•
•
•
Phase synchronization (PS),
Lag synchronization (LS),
Complete synchronization (CS), and
Generalized synchronization (GS)
(usually observed in coupled chaotic systems)
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Relationship between entropy and
self-organization
The relationship between entropy and self-organization
tries to relate organization to the 2nd Law of
Thermodynamics  order is a necessary result of farfrom-equilibrium (dissipative) systems trying to maximise
stress reduction. This suggests that the more complex
the organism then the more efficient it is at dissipating
potentials, a field of study sometimes called
'autocatakinetics' and related to what has been called
'The Law of Maximum Entropy Production'. Thus
organization does not 'violate' the 2nd Law (as often
claimed) but seems to be a direct result of it.
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What are dissipative systems ?
• Systems that use energy flow to maintain
their form are said to be dissipative (e.g.
living systems ).
• Such systems are generally open to their
environment.
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Biological signals
 Every living cell, organ, or organism generates
signals for internal and external communication.
 In-out relationship is generated by a biological
process (electrochemical, mechanical, biochemical
or hormonal).
 The received signal is usually very distorted by the
transmission channel in the body.
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Transport phenomena
(an elementary approach)
Current density (flux):
v
S
X
L
jX
jX = ρXv
ρX = X/V density
V = S·L volume
L = v·t
jXS = X/t
X = (mass, energy,
momentum, charge, ...)
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Transport phenomena
(an elementary approach)
• Continuity equation
∂tρX + div jX = 0
• Transport equation
jX = - αX grad ρX
αX(from kinetic theory) ~ vℓ
ℓ - mean free path
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Transport phenomena
(kinetic approach)
j1 = vρ(r + ℓ)
ℓ
ℓ
• The net flux through
the middle plane in
one direction is
j = (j2 – j1)/6
= - α gradρ
α = vℓ/6
j2 = vρ(r - ℓ)
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Transport phenomena
Mass, momentum, and energy transport
Diffusion(mass transport)
C(x - ℓ)
C(x + ℓ)
jD = v[C(x - ℓ) –C(x + ℓ)] / 6
Cv/6
ℓ
= v( - 2 ℓ ∂x C(x)) / 6
jD = - D ∂xC(x)
D=v ℓ/3
ℓ
x
C - concentration
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Transport phenomena
Mass, momentum, and energy transport
Heat transver (energy transport)
T(x - ℓ)
T(x + ℓ)
q = C v[Ek(x - ℓ) –Ek(x + ℓ)] / 6
Cv/ 6
ℓ
ℓ
x
c = ∂E/∂T = specific heat
= C v( - 2 ℓ ∂x Ek(x))/ 6
q = - κ ∂xT(x)
κ=vCℓc/3
C ( concentration ) = N / V
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Transport phenomena
Mass, momentum, and energy transport
Viscosity (momentum transport)
vy(x - ℓ)
vy(x + ℓ)
Πxy = C vm[vy(x - ℓ) –vy(x + ℓ)] / 6
y
Cv/6
ℓ
= C vm( - 2 ℓ ∂x vy(x)) / 6
Πxy = - η ∂xvy(x)
η = Cvm ℓ / 3
ℓ
x
C - concentration
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ENTROPY PRODUCTION
At the very core of the second law of thermodynamics we find the basic
distinction
between “reversible” and “irreversible processes” (1). This leads ultimately
to the introduction of entropy S and the formulation of the second
law of thermodynamics. The classical formulation due to Clausius refers to
isolated systems exchanging neither energy nor matter with the outside
world.
The second law then merely ascertains the existence of a function, the
entropy
S, which increases monotonically until it reaches its maximum at the state of
thermodynamic equilibrium,
(2.1)
It is easy to extend this formulation to systems which exchange energy and
matter with the outside world. (see fig. 2.1).
Fig. 2.1. The exchange of entropy between the outside and the inside.
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To extend thermodynamics to non-equilibrium processes we need an
explicit
expression for the entropy production.
Progress has been achieved along this
line by supposing that even outside equilibrium entropy depends only on the
same variables as at equilibrium. This is the assumption of “local”
equilibrium
(2). Once this assumption is accepted we obtain for P, the entropy
production per unit time,
(2.3) : dtSi = Σ Jα Fα
where the Jp are the rates of the various irreversible processes involved
(chemical
reactions, heat flow, diffusion. . .) and the F the corresponding generalized
266 Chemistry 1977
forces (affinities, gradients of temperature, of chemical potentials . . .). This
is the basic formula of macroscopic thermodynamics of irreversible
processes.
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