Attracting Manifolds Goes back to Tikhonov ( East ) and Levinson ( W est ) . Starting point: ODEs with scale separations: dt = 1 dx dy dt = f ( x, y ) ǫ g ( x, y ) Assumption: for every fixed x , the y - dynamics has a unique attracting fixed point , y= η ( x ) . Up to mild regularity assumptions on f , g , it can be shown that y ( t ) = η ( x ( t )) + O ( ε ) . To O ( ε ) corrections, x ( t ) is approximated by the solution X ( t ) of the reduced equation: dX dt = f ( X, η ( X ))
Attracting Manifolds Goes back to Tikhonov ( East ) and Levinson ( W est ) . Starting point: ODEs with scale separations: dt = 1 dx dy dt = f ( x, y ) ǫ g ( x, y ) Assumption: for every fixed x , the y - dynamics has a unique attracting fixed point , y= η ( x ) . Up to mild regularity assumptions on f , g , it can be shown that y ( t ) = η ( x ( t )) + O ( ε ) . To O ( ε ) corrections, x ( t ) is approximated by the solution X ( t ) of the reduced equation: dX dt = f ( X, η ( X )) Systems of class DD
y Actual trajectory η ( x ) y - dynamics “slaved” to x - dynamics attracting manifold x
Example : dx 1 dt = − x 2 − x 3 dx 2 dt = x 1 + 1 5 x 2 dx 3 dt = 1 5 + y − 5 x 3 dy dt = − y ǫ + x 1 x 3 ǫ
Example : dx 1 dt = − x 2 − x 3 dx 2 dt = x 1 + 1 5 x 2 dx 3 dt = 1 5 + y − 5 x 3 dy dt = − y ǫ + x 1 x 3 ǫ For fixed x y → x 1 x 3
Example : Reduced dynamics: dx 1 dt = − x 2 − x 3 dX 1 = − X 2 − X 3 dx 2 dt = x 1 + 1 dt 5 x 2 dX 2 = X 1 + 1 5 X 2 dx 3 dt = 1 dt 5 + y − 5 x 3 = 1 dX 3 5 + ( X 1 − 5) X 3 dy dt = − y ǫ + x 1 x 3 dt ǫ Rössler system For fixed x y → x 1 x 3
Example : Reduced dynamics: dx 1 dt = − x 2 − x 3 dX 1 = − X 2 − X 3 dx 2 dt = x 1 + 1 dt 5 x 2 dX 2 = X 1 + 1 5 X 2 dx 3 dt = 1 dt 5 + y − 5 x 3 = 1 dX 3 5 + ( X 1 − 5) X 3 dy dt = − y ǫ + x 1 x 3 dt ǫ Rössler system For fixed x y → x 1 x 3 Atrractor. ε =0.01 8 6 4 2 0 x 2 − 2 − 4 − 6 − 8 − 10 − 8 − 6 − 4 − 2 0 2 4 6 8 10 x 1 ǫ = 0 . 01
Example : Reduced dynamics: dx 1 dt = − x 2 − x 3 dX 1 = − X 2 − X 3 dx 2 dt = x 1 + 1 dt 5 x 2 dX 2 = X 1 + 1 5 X 2 dx 3 dt = 1 dt 5 + y − 5 x 3 = 1 dX 3 5 + ( X 1 − 5) X 3 dy dt = − y ǫ + x 1 x 3 dt ǫ Rössler system For fixed x y → x 1 x 3 Atrractor. ε =0.01 Attractor. 8 8 6 6 4 4 2 2 0 0 X 2 x 2 − 2 −2 − 4 −4 − 6 −6 − 8 −8 − 10 −10 − 8 − 6 − 4 − 2 0 2 4 6 8 10 −8 −6 −4 −2 0 2 4 6 8 10 x 1 X 1 ǫ = 0 . 01
Averaging First used in 3 - body celestial mechanics ( Lagrange 1788 ) . dt = 1 dx dy dt = f ( x, y ) ǫ g ( x, y )
Averaging First used in 3 - body celestial mechanics ( Lagrange 1788 ) . dt = 1 dx dy dt = f ( x, y ) ǫ g ( x, y ) Assumption : for fixed x , the y - dynamics are ergodic . Let denote the solution operator of the y - dynamics: ϕ t x ( y ) d ϕ 0 dt ϕ t x ( y ) = g ( x, ϕ t x ( y )) x ( y ) = y
Averaging First used in 3 - body celestial mechanics ( Lagrange 1788 ) . dt = 1 dx dy dt = f ( x, y ) ǫ g ( x, y ) Assumption : for fixed x , the y - dynamics are ergodic . Let denote the solution operator of the y - dynamics: ϕ t x ( y ) d ϕ 0 dt ϕ t x ( y ) = g ( x, ϕ t x ( y )) x ( y ) = y oung measure on Y : Ergodic dynamics induce a Y � T I A ( ϕ t µ x ( A ) = lim x ( y )) dt T →∞ 0 independent of y measure depends on x indicator function
Averaging ( cont. ) Anosov’s theorem states that x ( t ) converges uniformly on any bounded time interval to the solution X ( t ) of the averaged equation : dX � dt = f ( X, y ) µ X ( dy ) class DD Y
Averaging ( cont. ) Anosov’s theorem states that x ( t ) converges uniformly on any bounded time interval to the solution X ( t ) of the averaged equation : dX � dt = f ( X, y ) µ X ( dy ) class DD Y Comments :
Averaging ( cont. ) Anosov’s theorem states that x ( t ) converges uniformly on any bounded time interval to the solution X ( t ) of the averaged equation : dX � dt = f ( X, y ) µ X ( dy ) class DD Y Comments : • Extensive literature ( mostly Russian ) .
Averaging ( cont. ) Anosov’s theorem states that x ( t ) converges uniformly on any bounded time interval to the solution X ( t ) of the averaged equation : dX � dt = f ( X, y ) µ X ( dy ) class DD Y Comments : • Extensive literature ( mostly Russian ) . • Extension to cases where ergodicity fails on su ffi ciently small sets ( Arnold , Neistadt ) .
Averaging ( cont. ) Anosov’s theorem states that x ( t ) converges uniformly on any bounded time interval to the solution X ( t ) of the averaged equation : dX � dt = f ( X, y ) µ X ( dy ) class DD Y Comments : • Extensive literature ( mostly Russian ) . • Extension to cases where ergodicity fails on su ffi ciently small sets ( Arnold , Neistadt ) . • Extension to non - autonomous systems ( Artstein ) .
Averaging ( cont. ) Anosov’s theorem states that x ( t ) converges uniformly on any bounded time interval to the solution X ( t ) of the averaged equation : dX � dt = f ( X, y ) µ X ( dy ) class DD Y Comments : • Extensive literature ( mostly Russian ) . • Extension to cases where ergodicity fails on su ffi ciently small sets ( Arnold , Neistadt ) . • Extension to non - autonomous systems ( Artstein ) . • Extension to non - unique invariant measure ( di ff erential inclusions, Artstein ) .
Averaging ( cont. ) Anosov’s theorem states that x ( t ) converges uniformly on any bounded time interval to the solution X ( t ) of the averaged equation : dX � dt = f ( X, y ) µ X ( dy ) class DD Y Comments : • Extensive literature ( mostly Russian ) . • Extension to cases where ergodicity fails on su ffi ciently small sets ( Arnold , Neistadt ) . • Extension to non - autonomous systems ( Artstein ) . • Extension to non - unique invariant measure ( di ff erential inclusions, Artstein ) . • Invariant measure may depend on y ( 0 ) .
Application: Sti ff Hamiltonian Systems Ubiquitous in molecular systems: Hamiltonian systems with strong potential forces resuling in fast oscillatory motion around a sub - manifold, along with weaker forces responsible for conformational changes over longer timescales ( Rubin and Ungar, 1957, Neistadt 1984, Bornemann and Schuette 1997 ) . p 2 H ( z, p ) = 1 + V ( z ) + 1 � i ǫ 2 U ( z ) 2 2 m i i “soft” potential “sti fg ” potential The sti ff potential is minimal on a smooth submanifold M . Goal : approximate solution by a flow on M .
Example : a two - particle system constraining manifold y=0 2( p 2 + v 2 ) + V ( x ) + ω 2 ( x ) H ( x, p, y, v ) = 1 2 ǫ 2 y 2
Example : a two - particle system constraining manifold y=0 2( p 2 + v 2 ) + V ( x ) + ω 2 ( x ) H ( x, p, y, v ) = 1 2 ǫ 2 y 2 Equations of motion: dx dt = p dt = − V ′ ( x ) − ω ′ ( x ) dp 2 ǫ 2 y 2 dy dt = v ratio of small parameters dt = − ω ( x ) dv ǫ 2 y
Example : a two - particle system constraining manifold y=0 2( p 2 + v 2 ) + V ( x ) + ω 2 ( x ) H ( x, p, y, v ) = 1 2 ǫ 2 y 2 Equations of motion: dx dt = p dt = − V ′ ( x ) − ω ′ ( x ) dp 2 ǫ 2 y 2 dy dt = v ratio of small parameters dt = − ω ( x ) dv ǫ 2 y Assumption : energy E does not depend on ε , hence y → 0 as ε → 0 .
Example : a two - particle system constraining manifold y=0 2( p 2 + v 2 ) + V ( x ) + ω 2 ( x ) H ( x, p, y, v ) = 1 2 ǫ 2 y 2 Equations of motion: Naive solution: set y=0 . Wrong! dx dt = p dt = − V ′ ( x ) − ω ′ ( x ) dp 2 ǫ 2 y 2 dy dt = v ratio of small parameters dt = − ω ( x ) dv ǫ 2 y Assumption : energy E does not depend on ε , hence y → 0 as ε → 0 .
Example : a two - particle system constraining manifold y=0 2( p 2 + v 2 ) + V ( x ) + ω 2 ( x ) H ( x, p, y, v ) = 1 2 ǫ 2 y 2 Equations of motion: Naive solution: set y=0 . Wrong! dx dt = p dt = − V ′ ( x ) − ω ′ ( x ) dp 2 ǫ 2 y 2 This system is still not dy in the desired form of dt = v ratio of small parameters scale separation because dt = − ω ( x ) dv ǫ 2 y the “slow” ( x,p ) equations depend on ε . Assumption : energy E does not depend on ε , hence y → 0 as ε → 0 . Change variables: η = y / ε .
T ransformed system of equations: dx dt = p dt = − V ′ ( x ) − ω ′ ( x ) dp η 2 slow 2 dt = 1 d η fast ǫ v dt = − ω ( x ) dv η ǫ
T ransformed system of equations: dx The y - dynamics are ergodic : dt = p ( harmonic oscillator with dt = − V ′ ( x ) − ω ′ ( x ) dp η 2 x - dependent frequency ) . slow 2 dt = 1 d η fast ǫ v The invariant measure dt = − ω ( x ) dv η depends on the total energy ( i.e., ǫ on initial data of full system ) .
T ransformed system of equations: dx The y - dynamics are ergodic : dt = p ( harmonic oscillator with dt = − V ′ ( x ) − ω ′ ( x ) dp η 2 x - dependent frequency ) . slow 2 dt = 1 d η fast ǫ v The invariant measure dt = − ω ( x ) dv η depends on the total energy ( i.e., ǫ on initial data of full system ) . Applying the averaging principle ( here, a variation of Anosov’s theorem ) constant depending on dX dt = P initial data dP dt = − V ′ ( X ) − J ( ω 1 / 2 ( X )) ′ e fg ective ( Fixman ) potential ( non - trivial )
Stochastic Averaging A ( relatively ) strightforward generalizartion of the averaging method to stochastic systems: dx dt = f ( x, y ) dt = 1 ǫ g ( x, y ) + 1 dy √ ǫβ ( x, y ) dV dt
Stochastic Averaging A ( relatively ) strightforward generalizartion of the averaging method to stochastic systems: dx dt = f ( x, y ) dt = 1 ǫ g ( x, y ) + 1 dy √ ǫβ ( x, y ) dV dt If for fixed x the y - dynamics is ergodic with invariant measure μ x ( dy ) , then as ε → 0 , x ( t ) converges uniformly to X ( t ) : dX � dt = f ( X, y ) µ X ( dy ) class SD Y
Sketch of proof ( asymptotic analysis can be backed up by a limit theorem ) : dx dt = f ( x, y ) dt = 1 ǫ g ( x, y ) + 1 dy √ ǫβ ( x, y ) dV dt
Sketch of proof ( asymptotic analysis can be backed up by a limit theorem ) : dx dt = f ( x, y ) dt = 1 ǫ g ( x, y ) + 1 dy √ ǫβ ( x, y ) dV dt Step 1 : write corresponding Kolmogorov ( Fokker - Planck ) equation for φ ( x,y,t ) : ∂ 2 − 1 ∂ y ( g φ ) + 1 ∂φ ∂ t = − ∂ ∂ � β 2 φ � ∂ x ( f φ ) ∂ y 2 ǫ ǫ � �� � � �� � L 1 φ 1 ǫ L 0 φ
Sketch of proof ( asymptotic analysis can be backed up by a limit theorem ) : dx dt = f ( x, y ) dt = 1 ǫ g ( x, y ) + 1 dy √ ǫβ ( x, y ) dV dt Step 1 : write corresponding Kolmogorov ( Fokker - Planck ) equation for φ ( x,y,t ) : ∂ 2 − 1 ∂ y ( g φ ) + 1 ∂φ ∂ t = − ∂ ∂ � β 2 φ � ∂ x ( f φ ) ∂ y 2 ǫ ǫ � �� � � �� � L 1 φ 1 ǫ L 0 φ Step 2 : power series expansion: φ ( x, y, t ) = φ 0 ( x, y, t ) + ǫ φ 1 ( x, y, t ) + . . .
Step 3 : equate terms of same order: O ( 1/ ε ) terms : L 0 φ 0 = 0 ∂ 2 ∂ y ( g φ ) + 1 L 0 φ = − ∂ ∂ y 2 ( β 2 φ ) the generator of the y - dynamics 2 invariant distribution solution: φ 0 ( x, y, t ) = π ( x, t ) φ x eq ( y ) of y - dynamics
Step 3 : equate terms of same order: O ( 1/ ε ) terms : L 0 φ 0 = 0 ∂ 2 ∂ y ( g φ ) + 1 L 0 φ = − ∂ ∂ y 2 ( β 2 φ ) the generator of the y - dynamics 2 invariant distribution solution: φ 0 ( x, y, t ) = π ( x, t ) φ x eq ( y ) of y - dynamics L 0 φ 1 = ∂φ 0 O ( 1 ) terms : ∂ t − L 1 φ 0 L 1 φ = − ∂ ∂ x ( f φ ) the generator of the x - dynamics
Step 3 : equate terms of same order: O ( 1/ ε ) terms : L 0 φ 0 = 0 ∂ 2 ∂ y ( g φ ) + 1 L 0 φ = − ∂ ∂ y 2 ( β 2 φ ) the generator of the y - dynamics 2 invariant distribution solution: φ 0 ( x, y, t ) = π ( x, t ) φ x eq ( y ) of y - dynamics L 0 φ 1 = ∂φ 0 O ( 1 ) terms : ∂ t − L 1 φ 0 L 1 φ = − ∂ ∂ x ( f φ ) the generator of the x - dynamics Solvability condition : right - hand side orthogonal to the kernel of L 0* ( constant functions ) . Integrate over y : � �� �� ∂π ∂ t = − ∂ π ( x, t ) f ( x, y ) φ x eq ( y ) dy ∂ x
� �� �� ∂π ∂ t = − ∂ π ( x, t ) f ( x, y ) φ x eq ( y ) dy ∂ x Step 4 : W e identify this equation as the Liouville equation of the deterministic system dX � f ( X, y ) φ X dt = eq ( y ) dy ≡ F ( X )
� �� �� ∂π ∂ t = − ∂ π ( x, t ) f ( x, y ) φ x eq ( y ) dy ∂ x Step 4 : W e identify this equation as the Liouville equation of the deterministic system dX � f ( X, y ) φ X dt = eq ( y ) dy ≡ F ( X ) This asymptotic expansion can be made into a rigorous convergence proof ( e.g., through limit theorem for semi - groups ) .
Stochastic Limits A more subtle case of scale - separated system occurs for the following scaling: x - equation contains a “fast” term dt = 1 dx √ ǫ f 0 ( x, y ) + f 1 ( x, y ) dt = 1 ǫ g ( y ) + 1 dy √ ǫβ ( y ) dV dt y - equation independent of x ( skew - symmetric, not essential )
Stochastic Limits A more subtle case of scale - separated system occurs for the following scaling: x - equation contains a “fast” term dt = 1 dx √ ǫ f 0 ( x, y ) + f 1 ( x, y ) dt = 1 ǫ g ( y ) + 1 dy √ ǫβ ( y ) dV dt y - equation independent of x ( skew - symmetric, not essential ) The setting is such that f 0 ( x,y ) averages to zero under the invariant measure of the y - dynamics. A large term that averages to zero becomes, as ε → 0 , white noise .
Asymptotic expansion Step 1 : Switch to the Kolmogorov equation: ∂φ ∂ t = 1 ǫ L 0 φ + 1 √ ǫ L 1 φ + L 2 φ L 1 φ = − ∂ L 2 φ = − ∂ ∂ 2 ∂ y ( g φ ) + 1 L 0 φ = − ∂ ∂ y 2 ( β 2 φ ) ∂ x ( f 0 φ ) ∂ x ( f 1 φ ) 2
Asymptotic expansion Step 1 : Switch to the Kolmogorov equation: ∂φ ∂ t = 1 ǫ L 0 φ + 1 √ ǫ L 1 φ + L 2 φ L 1 φ = − ∂ L 2 φ = − ∂ ∂ 2 ∂ y ( g φ ) + 1 L 0 φ = − ∂ ∂ y 2 ( β 2 φ ) ∂ x ( f 0 φ ) ∂ x ( f 1 φ ) 2 Step 2 : Asymptotic series expansion: φ ( x, y, t ) = φ 0 ( x, y, t ) + √ ǫφ 1 ( x, y, t ) + ǫφ 2 ( x, y, t ) + . . .
Asymptotic expansion Step 1 : Switch to the Kolmogorov equation: ∂φ ∂ t = 1 ǫ L 0 φ + 1 √ ǫ L 1 φ + L 2 φ L 1 φ = − ∂ L 2 φ = − ∂ ∂ 2 ∂ y ( g φ ) + 1 L 0 φ = − ∂ ∂ y 2 ( β 2 φ ) ∂ x ( f 0 φ ) ∂ x ( f 1 φ ) 2 Step 2 : Asymptotic series expansion: φ ( x, y, t ) = φ 0 ( x, y, t ) + √ ǫφ 1 ( x, y, t ) + ǫφ 2 ( x, y, t ) + . . . Step 3 : Equate terms of same order O ( 1/ ε ) terms : L 0 φ 0 = 0 solution: φ 0 ( x, y, t ) = π ( x, t ) φ eq ( y )
L 0 φ 1 = − L 1 φ 0 O ( 1/ √ ε ) terms : Solvability condition requires that integral of RHS be zero. RHS = ∂ ∂ x [ f 0 ( x, y ) φ eq ( y ) π ( x, t )] This follows from the properties of f 0 . φ 1 = − L − 1 Solution: 0 L 1 φ 0
L 0 φ 1 = − L 1 φ 0 O ( 1/ √ ε ) terms : Solvability condition requires that integral of RHS be zero. RHS = ∂ ∂ x [ f 0 ( x, y ) φ eq ( y ) π ( x, t )] This follows from the properties of f 0 . φ 1 = − L − 1 Solution: 0 L 1 φ 0 L 0 φ 2 = ∂φ 0 O ( 1 ) terms : ∂ t − L 1 φ 1 − L 2 φ 0 Again, apply same solvability condition, and obtain an equation for the ( leading order ) marginal π( x,t ) : ∂π � � L 1 L − 1 ∂ t = − 0 L 1 φ eq ( y ) π ( x, t ) dy + L 2 φ eq ( y ) π ( x, t ) dy
Step 4 : identification of reduced problem: di fg usion drift ∂π � � L 1 L − 1 ∂ t = − 0 L 1 φ eq ( y ) π ( x, t ) dy + L 2 φ eq ( y ) π ( x, t ) dy operator in y first - order operator in x
Step 4 : identification of reduced problem: di fg usion drift ∂π � � L 1 L − 1 ∂ t = − 0 L 1 φ eq ( y ) π ( x, t ) dy + L 2 φ eq ( y ) π ( x, t ) dy operator in y first - order operator in x W e identify the equation for the marginal π( x,t ) as a Kolmogorov equation of a di fg usion process X ( t ) . The drift and di ff usion may be di ffi cult to evaluate analytically due to the need to invert L 0 .
f 0 ( x,y ) f 1 ( x,y ) dx dt = 1 Example : √ ǫ yx − λ x dy dt = − 1 ǫ y + 1 dV √ ǫ dt Ornstein - Uhlenbeck process
f 0 ( x,y ) f 1 ( x,y ) dx dt = 1 Example : √ ǫ yx − λ x dy dt = − 1 ǫ y + 1 dV √ ǫ dt Ornstein - Uhlenbeck process 1 ∂ 2 φ ∂ y ( y φ ) + 1 L 0 φ = ∂ √ π e − y 2 φ eq ( y ) = 2 ∂ y 2
f 0 ( x,y ) f 1 ( x,y ) dx dt = 1 Example : √ ǫ yx − λ x dy dt = − 1 ǫ y + 1 dV √ ǫ dt Ornstein - Uhlenbeck process 1 ∂ 2 φ ∂ y ( y φ ) + 1 L 0 φ = ∂ √ π e − y 2 φ eq ( y ) = 2 ∂ y 2 L 1 φ = − y ∂ ∂ x ( x φ ) L 2 φ = λ ∂ ∂ x ( x φ )
f 0 ( x,y ) f 1 ( x,y ) dx dt = 1 Example : √ ǫ yx − λ x dy dt = − 1 ǫ y + 1 dV √ ǫ dt Ornstein - Uhlenbeck process 1 ∂ 2 φ ∂ y ( y φ ) + 1 L 0 φ = ∂ √ π e − y 2 φ eq ( y ) = 2 ∂ y 2 L 1 φ = − y ∂ ∂ x ( x φ ) L 2 φ = λ ∂ ∂ x ( x φ ) Everything can be calculated analytically. ∂ 2 �� 1 � � ∂π ∂ t = − ∂ + 1 x 2 π � � 2 − λ x π ∂ x 2 ∂ x 2 Reduced Kolmogorov equation
f 0 ( x,y ) f 1 ( x,y ) dx dt = 1 Example : √ ǫ yx − λ x dy dt = − 1 ǫ y + 1 dV √ ǫ dt Ornstein - Uhlenbeck process 1 ∂ 2 φ ∂ y ( y φ ) + 1 L 0 φ = ∂ √ π e − y 2 φ eq ( y ) = 2 ∂ y 2 L 1 φ = − y ∂ ∂ x ( x φ ) L 2 φ = λ ∂ ∂ x ( x φ ) Everything can be calculated analytically. � 1 � ∂ 2 �� 1 � � dX X + X dU ∂π ∂ t = − ∂ + 1 x 2 π � � 2 − λ x π 2 − λ dt = ∂ x 2 ∂ x dt 2 Reduced Kolmogorov equation Reduced SDE
Original system: dx dt = 1 √ ǫ yx − λ x dy dt = − 1 ǫ y + 1 dV √ ǫ dt Reduced system: � 1 � dX X + X dU 2 − λ dt = dt class SS
Original system: Solution of reduced system: dx dt = 1 X ( t ) = X (0) exp [ − λ t + U ( t )] √ ǫ yx − λ x dy dt = − 1 ǫ y + 1 dV √ ǫ Properties ( a.s ) : dt Reduced system: λ > 0 t →∞ X ( t ) = 0 lim → � 1 � dX X + X dU λ = 0 lim sup t →∞ X ( t ) = ∞ → 2 − λ dt = dt λ < 0 t →∞ X ( t ) = ∞ lim → class SS
Original system: Solution of reduced system: dx dt = 1 X ( t ) = X (0) exp [ − λ t + U ( t )] √ ǫ yx − λ x dy dt = − 1 ǫ y + 1 dV √ ǫ Properties ( a.s ) : dt Reduced system: λ > 0 t →∞ X ( t ) = 0 lim → � 1 � dX X + X dU λ = 0 lim sup t →∞ X ( t ) = ∞ → 2 − λ dt = dt λ < 0 t →∞ X ( t ) = ∞ lim → class SS Numerical solution of log x ( t ) for ε =0.1 . λ = − 1 ε = 0.1 λ = 0 ε = 0.1 λ = 1 ε = 0.1 120 8 20 λ = - 1 λ =1 100 6 0 80 4 − 20 60 2 − 40 log(x) log(x) log(x) 40 0 − 60 20 − 2 − 80 λ =0 0 − 4 − 100 − 20 − 6 − 120 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 time time time
Example of Class DS x - equation contains a W e dealt with “fast” term systems of the dt = 1 dx √ ǫ f 0 ( x, y ) + f 1 ( x, y ) following form that yields dt = 1 ǫ g ( y ) + 1 dy √ ǫβ ( y ) dV dimension dt reduction of y - equation independent class SS . of x ( skew - symmetric, not essential )
Example of Class DS x - equation contains a W e dealt with “fast” term systems of the dt = 1 dx √ ǫ f 0 ( x, y ) + f 1 ( x, y ) following form that yields dt = 1 ǫ g ( y ) + 1 dy √ ǫβ ( y ) dV dimension dt reduction of y - equation independent class SS . of x ( skew - symmetric, not essential ) The same type of arguments remain valid if the y - dynamics are deterministic, but su ffi ciently - well mixing .
Example : fast term that 4 dx dt = x − x 3 + averages to zero 90 √ ǫ y 2 dt = 10 dy 1 y ( t ) satisfying ǫ ( y 2 − y 1 ) the ( accelerated ) dt = 1 dy 2 chaotic Lorenz ǫ (28 y 1 − y 2 − y 1 y 3 ) equations dt = 1 ǫ ( y 1 y 2 − 8 dy 3 3 y 3 )
Example : For ε → 0 , the slow component x ( t ) fast term that 4 dx dt = x − x 3 + averages to zero converges ( in Law ) 90 √ ǫ y 2 to the solution X ( t ) dt = 10 dy 1 y ( t ) satisfying ǫ ( y 2 − y 1 ) the ( accelerated ) of the SDE: dt = 1 dy 2 chaotic Lorenz ǫ (28 y 1 − y 2 − y 1 y 3 ) dX dt = X − X 3 + σ dU equations dt = 1 ǫ ( y 1 y 2 − 8 dy 3 3 y 3 ) dt ( σ =0.126 ) class DS .
Example : For ε → 0 , the slow component x ( t ) fast term that 4 dx dt = x − x 3 + averages to zero converges ( in Law ) 90 √ ǫ y 2 to the solution X ( t ) dt = 10 dy 1 y ( t ) satisfying ǫ ( y 2 − y 1 ) the ( accelerated ) of the SDE: dt = 1 dy 2 chaotic Lorenz ǫ (28 y 1 − y 2 − y 1 y 3 ) dX dt = X − X 3 + σ dU equations dt = 1 ǫ ( y 1 y 2 − 8 dy 3 3 y 3 ) dt ( σ =0.126 ) class DS . ε =0.001 Reduced equation describes ε 2 =0.001 1.4 noisy particle in quartic 1.2 potential. Equlibrium 1 Empirical Measure 0.8 distribution is bi - modal . 0.6 0.4 empirical distribution 0.2 0 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 x
Large Systems
Large Systems Another class of systems for which the reduced system can be derived rigorously as a limit of the full dynamics, is systems which many DOFs. The reduced system is obtained in the limit where the number of DOFs tends to infinity ( the “ thermodynamics limit ” ) . An instance of such systems are mechanical systems of heat baths . ( Will be addressed in detail in tomorrow’s lecture ) . Also systems of class DS .
Birth - Death Systems Chemical reactions are commonly modeled by stochastic birth - death systems. The model : There are m species with populations x= ( x 1 ,x 2 ,...,x m ) . There are n reactions with rates h i ( x ) and stoichiometry numbers ν ij . Can easily be simulated by the Gillespie algorithm ( generative simulation of cont. - time Markov chains )
The Gillespie algorithm :
The Gillespie algorithm : • Initialize x i ( 0 ) for i=1,..., m .
The Gillespie algorithm : • Initialize x i ( 0 ) for i=1,..., m . • Compute the reaction rates r j =h j ( x ( 0 )) for j=1,..., n .
The Gillespie algorithm : • Initialize x i ( 0 ) for i=1,..., m . • Compute the reaction rates r j =h j ( x ( 0 )) for j=1,..., n . • Set the total rate r =r 1 +...+r n .
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