Local time path integrals and their application to Lévy random walks Václav Zatloukal (www.zatlovac.eu) Faculty of Nuclear Sciences and Physical Engineering Czech Technical University in Prague talk given at Path Integration in Complex Dynamical Systems 8 February 27 Leiden V. Zatloukal Local time path integrals /33
Motivation TIME ZONES: V. Zatloukal Local time path integrals 2/33
Motivation TIME ZONES: NOT INCLUDED IN THIS PRESENTATION V. Zatloukal Local time path integrals 2/33
Motivation Local time of a stochastic process: xhtl l=2 l=.5 t - l= -2 V. Zatloukal Local time path integrals 3/33
Motivation Local time of a stochastic process: xhtl x x x l=2 l=.5 t.4.8.2 LHxL.5.5 2 2.5 LHxL 2 3 4 5 6 7 LHxL - - - - l= -2-2 -2-2 l=2 l=.5 l= Stochastic trajectory x( )! Local time profile L(x) V. Zatloukal Local time path integrals 3/33
Outline Preliminaries: di usion equation and path integral Introduction of local time of a stochastic process Correlation functions and functionals of the local time Time-independent systems and the resolvent method Local time of Lévy random walks (Lévy flights) Local-time representation for Gaussian path integrals In collaboration with Petr Jizba (Czech Technical University in Prague) and Angel Alastuey (ENS Lyon). V. Zatloukal Local time path integrals 4/33
Di usion equation and path integral Di usion (or heat, or Fokker-Planck) equation [@ t + H( i@ x, x, t)] P(x, t) = () where H(p, x, t) is a generic Hamiltonian, and x 2 R. [~ =] Solution for initial condition P(x, t a )= (x x a ) represented by phase-space path integral (x b t b x a t a ) P(x b, t b )= x(tz b )=x b x(t a )=x a Dx Z Dp 2 ea[p,x] (2) where the action A[p, x] = Z tb t a d ip( )ẋ( ) H(p( ), x( ), ) (3) V. Zatloukal Local time path integrals 5/33
Di usion equation and path integral We will gradually specify the Hamiltonian: H(p, x, t) generic, possibly time-dependent [ H(p, x) time-independent [ H(p) time and position-independent [ H (p) =D (p 2 ) /2 Lévy Hamiltonian In the end, special attention to H(p, x) = p2 2M + V (x) usual quantum Hamiltonian V. Zatloukal Local time path integrals 6/33
Di usion equation and path integral Significance of the heat equation: Di usion processes (or continuous random walks) P(x, t) is the probability of transition to point x after time t Quantum mechanics: by rotation to imaginary times t! it P are transition amplitudes Quantum statistical physics: identifying t = ~, where = k B T P are matrix elements of the equilibrium density matrix... In addition, corresponding path integrals describe: Polymers or other line-like objects Stock prices on financial markets... V. Zatloukal Local time path integrals 7/33
Local time of a stochastic process xhtl x x x l=2 l=.5 t.4.8.2 LHxL.5.5 2 2.5 LHxL 2 3 4 5 6 7 LHxL - - - - l= -2-2 -2-2 l=2 l=.5 l= For stochastic trajectory x( ) define L(x; t a, t b, x( )) = Z tb t a d (x x( )) (4) P. Lévy, Sur certains processus stochastiques homogènes, Compos. Math. 7, 283(939). V. Zatloukal Local time path integrals 8/33
Local time of a stochastic process L is functional of x( ) and function of x: L(x) R R L(x)dx = t b t a L(x) has compact support L(x) is continuous L(x)... trajectories of a new stochastic process V. Zatloukal Local time path integrals 9/33
Correlation functions of local time hl(x )...L(x n )i = Z x(t b )=x b x(t a )=x a Dx = X 2S n Z t a <t <...<t n <t b dt...dt n Z Dp 2 ea[p,x] L(x )...L(x n ) ny (x (k+) t k+ x (k) t k ) (5) where are permutations of indices {,...,n}, with () = and (n + ) = n +, and we have denoted x = x a, t = t a, x n+ = x b, t n+ = t b. k= V. Zatloukal Local time path integrals / 33
Generic functionals of local time L(x )...L(x n ) is an example of functional of the local time F [L(x)]. In general, hf [L(x)]i = = F x(tz b )=x b x(t a )=x a Dx apple U(x) Z Dp 2 ea[p,x] F [L(x)] x(tz b )=x b x(t a )=x a Dx Z Dp 2 ea U[p,x] U= (6) where A U [p, x] corresponds to H U (p, x, t) =H(p, x, t)+u(x). We have employed the key identity Z tb t a U(x( )) d = Z U(x)L(x) dx (7) V. Zatloukal Local time path integrals / 33
Time-independent systems and the resolvent method Assume H(p, x) [ independent of time t ] ) (x b t b x a t a )=hx b e (t b t a )Ĥ x a i! take t a =, t b t from now on. Laplace representation hf [L(x)]i E = R dt e te hf [L(x)]i Correlation functions: hl(x )...L(x n )i E = X 2S n ny k= R(x (k), x (k+), E) (8) [ e.g. hl(x)i E = R(x a, x)r(x, x b ) ] Here R(x, x, E) hx (Ĥ + E) xi (9) denotes matrix elements of the resolvent operator ˆR(E) =(Ĥ E). V. Zatloukal Local time path integrals 2 / 33
Time-independent systems and the resolvent method Generic functionals: apple hf [L(x)]i E = F R U (x a, x b, E) U(x) U= where ˆR U (E) =(ĤU E) is the resolvent corresponding to the extended Hamiltonian H U = H + U., () One-point distribution: take U(x )=u (x x)! U(x) = @ @u... ) R U (x a, x b )=R(x a, x b ) u R(x a, x)r(x, x b ) +ur(x, x) (L L(x)) E = (L)R(x a, x)r(x, x b ) R(x, x) 2 e apple + (L) R(x a, x b ) L R(x,x) R(x a, x)r(x, x b ) R(x, x) () (2) V. Zatloukal Local time path integrals 3 / 33
*Multipointresolvents Resolvent identity for reads (ĤU + E) =(Ĥ + E) (ĤU + E) U(ˆx)(Ĥ + E) (3) U(x) = R U (x a, x b )=R(x a, x b ) Setting x a = x k for k =,...,n, where M jk R U (x j, x b )= nx u j (x x j ) (4) j= nx k= nx u j R(x a, x j )R U (x j, x b ) (5) j= are elements of the inverse of the matrix M jk R(x k, x b ) (6) M jk = jk + u k R(x j, x k ) (7) V. Zatloukal Local time path integrals 4 / 33
Distribution functions and moments of local time Invert the Laplace transform to return to the time domain: h...i E!h...i Moments of local time: µ(x,...,x n )= hl(x )...L(x n )i hi (8) n-point distribution functions of local time: Q n j= W (L,...,L n ; x,...,x n )= (L j L(x j )) hi (9) The normalization factor hi equals (x b t b x a t a ) for averaging over paths with fixed initial point x a and final point x b. Averaging over paths with arbitrary x b : h...i R dx b h...i!hi,µ, W,... V. Zatloukal Local time path integrals 5 / 33
Lévy random walks Hamiltonians H(p) only dependent on momentum form now on: P(x, t) (x b t x a ) = Z dp 2 e th(p) e ipx, x = x b x a (2) For the Lévy Hamiltonian H (p) =D (p 2 ) /2 [ we consider 2 [, 2] ]! symmetric Lévy stable distribution Heavy tails for For =2Gaussian: P =2 (x, t) = e x 2 4D 2 t p 4 D2 t <2: P (x, t) x! c x +.3.25.2 P l HxL l=2 l=.5 l= For =Cauchy(-Lorentz): P = (x, t) = D t (D t) 2 + x 2-4 -2 2 4.5..5 x V. Zatloukal Local time path integrals 6 / 33
Lévy stable distributions Stability: X Levy, X 2 Levy ) X + X 2 Levy Characteristic function: h exp ipµ cp i sgn(p) i, = ( tan( /2) 6= (2/ ) log p =...tail power ( x ), c...width, µ...shift of origin,...asymmetry Generalized central limit theorem: i.i.d. X,...,X n distribution with possibly infinite variance lim n! X +...+ X n N n Levy Applications in financial markets (fluctuation of stock prices), physics (e.g. François Bardou et al., Lévy Statistics and Laser Cooling: How Rare Events Bring Atoms to Rest, CUP(22).,...),... V. Zatloukal Local time path integrals 7 / 33
Lévy random walks xhtl x x x l=2 l=.5 t.4.8.2 LHxL.5.5 2 2.5 LHxL 2 3 4 5 6 7 LHxL - - - - l= -2-2 -2-2 l=2 l=.5 l= For decreasing : Longer jumps (Lévy flights) Separation of local time L(x) into peaks V. Zatloukal Local time path integrals 8 / 33
Resolvent of Lévy Hamiltonian R (x, x, E) = Z dp 2... Linnik (or geometric stable) distribution e ip(x x) D (p 2 ) /2 + E (2) For =2 Gaussian resolvent: R l HxL.8 R 2 (x, E) = e p E/D2 x p 4D2 E.6 l=2 l=.5 For = Cauchy resolvent: apple sin E x D R (x, E) = D + cos E x D si E x D ci E x D -4-2 2 4.4.2 l= divergent for = at x = x V. Zatloukal Local time path integrals 9 / 33
One-point distributions of local time at the origin ) x b = x a : (L L(x a )) E = e L R () =exp h LE i, D / sin W (L; x a )= = (D t)/ ( ) (L Z hi L(xa )) de e Et exp h LE cos i sin hle sin i (22) Gaussian case: For W =2 (L; x a )= 2D 2L t =(Cauchy case): singularity R () = + e D 2L 2 /t! flattening of the distribution W l HLL..8.6.4.2 l=2 l=.5 l=. 2 3 4 5 L V. Zatloukal Local time path integrals 2 / 33
One-point distributions of local time at the origin 2) free x b : (L L(x a )) E = ER () e L R () = E / exp h LE i (L W (L; x a )= Z = de e Et E / L(xa )) hi (23) h exp LE cos i sin hle sin + i W l * HLL Gaussian case: W =2 (L; x a)= r 4D2 t e D 2L 2 /t..8.6.4 l=2 l=.5 l=..2 2 3 4 5 L V. Zatloukal Local time path integrals 2 / 33
*One-pointdistributionofBrownianlocaltime Gaussian case =2(Brownian motion): generic x b and x W 2 (L; x) = (L) x a x + x x b +2D 2 L apple t ( xa x + x x b +2D 2 L) 2 (x b x a ) 2 exp 4D 2 t apple ( xa x + x x b ) 2 (x b x a ) 2 + (L) exp 4D 2 t (24) r apple W2 4D2 (L; x) = (L) t exp ( xa x +2D 2 L) 2 4D 2 t + (L) erf apple xa x p 4D2 t (25) The -term vanishes for x between x a and x b. V. Zatloukal Local time path integrals 22 / 33
First moments of local time [ x b = x a ] [free x b ] m l HxL m l * HxL.5 l=2.5 l=2 l=.5 l=.5. l=. l=.5.5-3 -2-2 3 x -3-2 - 2 3 x µ =2 (x) = s (x t b xa) 2 e 4D 2 t 4D 2 erfc " xa x + x x b p 4D2 t # µ =2 (x) = s t D 2 e (xa x) 2 4D 2 t x a x 2D 2 " # xa x erfc p 4D2 t µ = (x) = n 2 D 2 h t (D t) 2 +(xa+x b 2x) 2i h 2(x x a ) h +2(x b x) (D t) 2 +(x b x) 2 (x x a ) 2i arctan D t x b x h +D t (D t) 2 +(x x a ) 2 +(x b x) 2i ln (D t) 2 +(x x a ) 2 (x b x) 2i arctan D t x xa (x xa) 2 (x b x) h 2 (D t) 2 +(x xa) 2ih (D t) 2 +(x b x) 2 i ) ) µ = (x) = " ln + 2 D x D t x a 2 # V. Zatloukal Local time path integrals 23 / 33
*SecondmomentsofBrownianlocaltime ) fixed x b µ 2 (x, x 2 )= t 2D 2 +( 2 $ 2 ) apple 2 exp 2 (x b x a ) 2 4D 2 t p apple 2 p 4D2 t erfc 2 p 4D2 t apple (xb x a ) 2 exp 4D 2 t where 2 x a x + x x 2 + x 2 x b, 2 x a x 2 + x 2 x + x x b 2) free x b µ 2 (x, x 2 )= 4D 2 p apple t p 2 ( exp 2 ) 2 D2 4D 2 t + apple ( 2 ) 2 2D 2 + t erfc apple 2 p 4D2 t +( 2 $ 2) where 2 x a x + x x 2, 2 x a x 2 + x 2 x V. Zatloukal Local time path integrals 24 / 33
Path integrals in quantum statistical physics From now on H(p, x) = p2 2M + V (x) (26) Motivation in quantum statistical physics: Gibbs operator: e! partition function: Tr (e! thermal density matrix: e Ĥ Ĥ ), =/k B T Ĥ /Tr (e Ĥ ) (x a, x b, ) hx b e Ĥ x a i... Heat kernel for di usion generated by H, with time variable : @ + H( i~@x, x) (x a, x, )= (27) V. Zatloukal Local time path integrals 25 / 33
Path integrals in quantum statistical physics ) Feynman path-integral representation For (x a, x b, )= Z x( ~)=x b x()=x a Dx( )exp!... high temperatures: ~ Z ~ d apple M 2 ẋ 2 ( )+V (x( )) (28) V (x( )) = V (x a )+V (x a )(x( ) x a )+...) e V (x a ) p 2 ~ 2 /M P. Jizba and V. Zatloukal, Path-integral approach to the Wigner-Kirkwood expansion, Phys. Rev. E 89, 235(24) 2) Spectral representation (x a, x b, )= X n e E n n(x a ) n (x b ) where Ĥ n i = E n n i (29) For!... low temperatures: e E gs gs(x a ) gs (x b ) V. Zatloukal Local time path integrals 26 / 33
Local-time path integral 3) Local-time representation? path integral over local time profiles! s = / ~ L(x) = Z ~ d (x x( )) (3) small large V. Zatloukal Local time path integrals 27 / 33
Local-time path integral (x a, x b, )= Z x( ~)=xb x()=x a Dx( )exp ( ~ Z ~ M 2 ẋ 2 ( )d ~ Z R V (X )L(X )dx ) + change of variables: x( )! L(x) (x a, x b, )= Z DL(x)W [L(x); + ~, x a, x b ]exp Z R L(x)V (x)dx where L(x) and W [L(x)] = if R R L(x)dx 6= ~! identify the weights W [L(x)] V. Zatloukal Local time path integrals 28 / 33
Local-time path integral: glimpses of the derivation Di usion equation in Laplace picture: E + H( i~@x, x) e (x a, x, E) = (x a x) Quantum-field-theoretic representation: e (x a, x b, E) =2 R D (x) (x a ) (x b )e h E+Ĥ i / R D (x)e h E+Ĥ i Replica trick: a/b =lim D! ab D R PD o 2 e =lim D! D D D (x) (x a ) (x b )expn = h E + Ĥ i Spherical coordinates in -space: radial part = q ~ ~ Inverse Laplace transform ) Local-time representation of (x a, x b, ) V. Zatloukal Local time path integrals 29 / 33
Local-time path integral (x a, x b, ) hx b e Z (X+ )= + (X )= Ĥ x a i = D (x) lim lim X ±!± Z X+ X D! 2 D 2 2 (x)dx lim ( +) ±!! D 2 (x a ) (x b )exp{ A [ (x)]} where (x) 2 (x) $ L(x)/~ The action of a radial harmonic oscillator ($ Bessel process indexed by x) A [ (x)] Z X+ X dx apple ~ 2 2M (x) 2 + V (x) 2 (x)+ M ~ 2 (x) 2 2 (x) (3) V. Zatloukal Local time path integrals 3 / 33
Local-time path integral at low temperatures Rescaling! p : A [ p (x)] = Z X+ X dx apple ~ 2 2M (x) 2 + V (x) 2 (x)+ M ~ 2 (x) 2 2 2 (x)!: Saddle-point approximation of the path integral (! neglect the last term) $ Minimization of the functional: Z X+ X dx apple ~ 2 2M (x) 2 + V (x) 2 (x) = h Ĥ i under the constraint h i = ) Rayleigh-Ritz variational principle for the ground state V. Zatloukal Local time path integrals 3 / 33
Generic functionals of the local time Average value of generic functional F [L(x)]: Z x( ~)=xb x()=x a = lim Dx( )F [L(X )] exp lim X ±!± D! Z (X+ )= + (X )= 2 D 2 ( ~ Z ~ lim ( +) ±! D (x)f [~ 2 (x)] d D 2 apple M 2 ẋ 2 ( )+V (x( )) Z X+ X 2 (x)dx! ) (x a ) (x b )e A [ (x)] V. Zatloukal Local time path integrals 32 / 33
Summary We discussed local times of stochastic processes for generic Hamiltonian Calculated moments and distributions of local time In particular, for Lévy random walks For Gaussian path integrals derived local-time path-integral representation xhtl x x x l=2 l=.5 t.4.8.2 LHxL.5.5 2 2.5 LHxL 2 3 4 5 6 7 LHxL - - - - l= -2-2 l=2-2 l=.5-2 l= P. Jizba and VZ, Local-time representation of path integrals, Phys. Rev. E 92, 6237(25)[arXiv:56.888]. VZ, Local time of Lévy random walks: a path integral approach, [arxiv:72.2488]. V. Zatloukal Local time path integrals 33 / 33