Introduction to Large Deviation Principle and its applicationsbicmr.pku.edu.cn/~gehao/Chinese...
Transcript of Introduction to Large Deviation Principle and its applicationsbicmr.pku.edu.cn/~gehao/Chinese...
Introduction to Large Deviation Principle and its applications
Hao Ge1Biodynamic Optical Imaging Center (BIOPIC) 2Beijing International Center for Mathematical
Research (BICMR)Peking University, China
Mathematical foundation of statistical mechanics
Mathematical analysis Classic mechanics
Statistical mechanics?
Mathematical structure of statistical mechanics?
Only math can be applied beyond physics!
Law of Large Number
,...,...,, 21 nXXX Independent, identical distributed random variables
nn XXXS +⋅⋅⋅++= 21
+∞→=→ nXnSn ,1µ
0,,0 >∀+∞→→
>− εεµ n
nSP n
Strong version
Weak version
Central Limit Theorem
( )122
1
var,21)(
0,,)(
2
Xey
ndyyxn
nSP
y
xn
==
>∀+∞→→
<
−
−
∞−∫
σπ
φ
εφσµ
( ) ( ) 22 2/
21 σµ
πσnnS
nne
nSp −−≈
Central Limit Theorem
Variance = nσ2
Sn
µn
P(Sn)
Cramer’s theorem
)(~ xnIn exnSP −
=
{ })(sup)( θθθ
CxxI −=
∫ == )(log)( 1 xXPeC xθθ Cumulant generating function
Rate function
{ } 0)(inf)( == xIIx
µ 0)(' =µI
21)(''
σµ =I ( ) 1,
21)( 2
2 <<−−≈ µµσ
xxxI
)(log1 xIxnSP
nn −→
=
Central limit theorem
Sanov theorem
Empirical distribution:
∑=
=n
ijXjn in
L1
,,1 δ
( ) ∑=
=M
i i
iiM p
lllllI1
21 ln,...,,
jpjXP == )( 1
( ) ),...,(,
1~,...,2,1, MllnIjjn eMjlLP −==
Relative entropy
Gibbs conditioning
{Xi | i=1,2,…,n} i.i.d. with probability density fpriori
Under the conditiong(X1) + g(X2) +…+ g(Xn) = α = na
What is the asymptotic posterior distribution?
µ≠a
( )naXgXgXgXPxf nnposterior =+++=∞→
)(...)()(|lim)( 211
Minimum relative entropy principle
Xi
Xj
Now, if n is very large, then a is essentially the expected value of the posterior distribution for each Xi!Minimum relative entropy:
dxxfxf
xfpriori
posteriorposterior )(
)(ln)(∫
subjected to: .)()( adxxfxg posterior =∫)()()( xg
prioriposterior exfxf β−∝
Varadhan theorem
Generalization of Laplace method
{ }.)()(supln1lim)( )( xIxfen
fx
Anf
nn −==
∞→λ
( ) )(~ xnIn exAP −=If
Then
Varadhan, SRS (1966). "Asymptotic probabilities and differential equations". Communications on Pure and Applied Mathematics 19 (3): 261–286
(Large deviation)
Contraction principle
( ) )(~ xnIn exAP −=If
Then ( )( ) )(~ ynJn eyAfP −=
)(inf)()(
xIyJyxf =
=
== ∑=
xillllIxIM
iiM
121 :),...,,(inf)(E.g.
Gartner-Ellis theoremHugo Touchette, Phys. Rep. 478, 1-69 (2009)Ellis, Entropy, Large Deviation and Statistical Mechanics. (1984)
nnkA
ne
nk ln1lim)(
∞→=λIf
exists and is differentiable
Then ( ) )(~ xnIn exAP −=
{ }.)(sup)( kkxxIk
λ−=
Statistical mechanics
{ }.)(sup)( kkxxIk
λ−=Legendre–Fenchel transformation
)()( ** kxkxI λ−= )(' *kx λ=
entropy Free energy/temperature energy 1/temperature
Hugo Touchette, Phys. Rep. 478, 1-69 (2009)Ellis, Entropy, Large Deviation and Statistical Mechanics. (1984)
)(~ xnIn exnhP −
=
energy
Markov processDonsker and Varadhan: Asymptotic evaluation of certain Markov process expectations for large time. I, Communications on Pure and Applied Mathematics 28 (1975), pp. 1–47; part II, 28 (1975), pp. 279–301; part III, 29 (1976), pp. 389–461; part IV, 36 (1983), pp. 183–212 Dembo and Zeitouni, Large Deviations Techniques and Applications (2009)
{ },...,...,, 21 nXXX Finite state Markov chain
( ) ijkk piXjXP ===+ |1 ∑=
=n
ikn Xf
nA
1
)(1
)(),( jfijepjiP λ
λ =( ) )(~ xnIn exAP −=
( ){ }λλ
ρλ PxxI logsup)( −= Perron-Frobenius eigenvalue
Freidlin-Wentzell theoryFreidlin and Wentzell: Random Perturbation of Dynamical System. (1983)
Ge and Qian: PRL (2009), JRSI (2011), Chaos (2012)
( ) ttt Xb
dtdX η+= Aststt εδηηη ,,0 ==
{ }( ) ( )( ) ( ) ( )( )∫=
− −−=≤≤T
ssss
Tsss dsxbxxAxbxTsxI
0
'1'21
0:
{ }( ) { }( )TssxI
ss eTsxXP≤≤−
≤≤=0:1
~0: ε
Strongly dependent on the Gaussian feature of the white noise
Freidlin-Wentzell theory
A nonequilibrium generalization of Kramer’s rate theory
{ }( ) ( )( ) ( ) ( )( )∫=
− −−=≤≤T
ssss
Tsss dsxbxxAxbxTsxI
0
'1'21
0:
O1
O2
Saddle
Φ
Transition rate
( ) { }( )TsxIx sxTxOx≤≤=Φ
==0:inf
,10
ε∆Φ
−
∝ ekk 0
( )0
)(≤
Φdt
txd( )tt xb
dtdx
=
Quasipotential/landscape
Emergent dynamic landscape
)(loglim)(0
xpx ssalso
εεεφ
→−=
Global minimum
Local minimum
Maximum: the barrier
Stable fixed points of deterministic models
Unstable fixed point of deterministic models
( )tt xb
dtdx
=
Relative stability of stable steady states
Many nonlinear dynamical systems have multiple, locally stable steady states.Is one attractor more “important” than another?
( ) .0,1exp)( →
−≈ εφ
εε xxpss
The most important steady state when V is large would be the global minimum of dynamic landscape.
Maxwell construction
),( θxbdtdx
=θ *
φ (x,θ )
θ
Steady States x*
x
Global minimum abruptly transferred.
Ge and Qian: PRL (2009), JRSI (2011)
Nonequilibrium phase transition
( ) ( ){ }kkck
φλλ −= sup ( ) ( ){ }kckxxck
−= sup*
Ge and Qian: PRL (2009), JRSI (2011)
alternative attractor
2: fluctuating in local attractor, waiting
1: relaxation process
3: abrupt transition via barrier-crossing
The uphill dynamics is the rare event, related to phenotype switching, punctuated transition in evolution, et al.
Dynamics of bistable systems
Intra-attractorial dynamics
Inter-attractorial dynamics
Local-global confliction
Global landscape: stationary distribution
Just cut and glue on the local landscapes (non-derivative point).
The emergent Markovian jumping process being nonequilibrium is equivalent to the discontinuity of the local landscapes (time symmetry breaking).
Dynamic on a ring as an example.
Local landscapes
Kramers’ rate formula
Ge and Qian: Chaos (2012)
Chemical master equation
Chemical Master Equation
Gillespie algorithm Langevin dynamics
Fokker-Planck equation
Gillespie algorithm (GA) is really an equation that describes the dynamic trajectory of chemical master equation.
Trajectory view
Probability distribution view
Discrete Markov Chain
Diffusion process
Also isomorphic to self-regulating gene system
Self-regulating gene
Phosphorylation-dephosphorylation cycle
With positive feedback
The dimer case χ=2
E E*
K*
ATP ADP
P
Pi
a1
a-1
a2
a-2
K and K* are inactive and active forms of a kinase. E* is the phosphorylated E, a signaling molecule. Usually E∗ is functionally active, i.e., “turned-on”.
E E*
K
P
2E*0E* 1E* 3E* … (N-1)E* NE*
Chemical master equation Representation
v1
w1
v2
w2
v0
w0
.))2)(1((
),1)()2)(1((
2
2
nnnV
w
nNnnV
v
n
n
δε
β
δα
+−−+=
+−+−−=
Emergent dynamic landscape
)(uφ
Global minimum
Local minimum
Maximum: the barrier
Stable fixed points of deterministic models
Unstable fixed point of deterministic models
( )( )
.,
,exp)(
+∞→=
−∝
VVnu
uVnpssV φ
Maxwell construction
),( θubdtdu
=θ *
φ (u,θ )
θ
Steady States u*
u
Global minimum abruptly transferred.
Ge and Qian: PRL (2009), JRSI (2011)
Kramers’ theory for CME
., 12211221
→→ −→
−→ ∝∝ VHVH eTeT
21→H
The switching time between attractors:
12→H
The barrier H here may not be the same as the barrier in the global landscape φ(u) for high dimensional multistable cases.
A B
discrete stochastic model among attractors
ny
nx
chemical master equation cy
cx
A
B
fast nonlinear differential equations
appropriate reaction coordinate
A Bprob
abil
ity
emergent slow stochastic dynamics and landscape
(a) (b)
(c)(d)
Three time scalesFixed finite molecule numbers
Stochastic
Stochastic
Deterministic
Ge and Qian: PRL (2009), JRSI (2011)
Possible Chemical basis of epi-genetics:
Could this be a chemical definition for epi-genetics inheritance: Exactly same environment setting and gene, different internal biochemical states (i.e., concentrations and fluxes)?
It is likely that the code for epi-genetic inheritance is distributed.
Could it be readily inherited during the process of cell volume change and division?
Acknowledgement
Prof. Hong Qian
University of WashingtonDepartment of Applied Mathematics
My collaborator: