Perturbation spreading in many particle systems: a random walk approach Peter Hänggi, Institut für Physik, Universität Augsburg V. Zaburdaev, S. Denisov and P. Hänggi Perturbation spreading in many particle systems: a random walk approach Phys. Rev. Lett. 106, 180601 (2011);
Transport in 1D systems Transport in Hamiltonian systems is anomalous [Zaslavsky (2002)] Casual cone: general feature of many-body systems [Lieb & Robinson (1972), Marchioro et al. (1978)] What is inside the cone? What is on the front? What about correlations? [V. Zaburdaev, S. Denisov, and P. Hanggi PRL 106 180601 (2011)]
Two examples N X H total ð f x i ; p i g Þ ¼ H i ; where H i ¼ H ð x i ; x i � 1 ; x i þ 1 ; p i Þ is i th particle. At the time t ¼ i ¼ 1 Hard Point Gas (HPG) m M particle, " ¼ h m i v 2 M/m = r i i = 2 , m i ¼ m or M , FPU b chain H i = 1 i + 1 2( x i +1 − x i ) 2 + β 2 p 2 4 ( x i +1 − x i ) 4 is " ¼ H total =N .
Energy perturbation by 4 E ð i; t Þ ¼ H p i ð t Þ � H i ð t Þ . i.e., � N i ¼ 1 4 E ð i; t Þ ¼ E p . a n t ifi e d wi t h a norm a liz e n % ð i; t Þ ¼ 4 E ð i; t Þ =E p , 1 0.5 1e-02 ni ca l a v e r a g e . # # 3 # 6 N = 1 N = 10 N = 10 ∞ � σ 2 ( t ) = i 2 ̺ ( i, t ) 0.5 0.25 −∞ 5e-03 � E(i,t=400) 0 0 Features: non-negative -0.5 -0.25 0e+00 σ 2 ( t ) ∝ t α anomalous: -1 -0.5 casual cone 0 1000 2000 0 1000 2000 0 1000 2000 i i i FPU
Random walk Levy walks PDF of flight times: [Geisel (1985), Klafter (1982)] 1 ψ ( τ ) = γ γ > 0 (1 + τ / τ 0 ) 1+ γ τ 0 | x | = v 0 τ
Levy walk 1 ψ ( τ ) = γ (1 + τ / τ 0 ) 1+ γ τ 0 P ( x, t ) ∝ 1 te − x 2 /t √ x 2 � � γ > 2; < ∞ x t ∝ | x | 1+ γ x 2 � � 1 < γ < 2; = ∞ ; x 0 < γ < 1; � | x | � = ∞ ; x
Random walk PDF of flight times: ψ ( τ ) ∝ ( τ / τ 0 ) − γ − 1 γ > 0 0 10 σ 2 ( t ) ∝ t 3 − γ superdiffusion: 1 < γ < 2 1 x � � scaling: and u = t ′ /t , P ( x, t ′ ) ≃ Ku 1 / γ P Ku 1 / γ , t P(x,t=100) -2 erence between 10 | x | = v 0 t ballistic peaks: -4 10 0 50 100 150 x
Random walk PDF of flight times: 1 0.5 1e-02 3 # # 6 # N = 1 N = 10 N = 10 ψ ( τ ) ∝ ( τ / τ 0 ) − γ − 1 γ > 0 0.5 0.25 5e-03 � E(i,t=400) 0 0 0 10 σ 2 ( t ) ∝ t 3 − γ superdiffusion: 1 < γ < 2 1 x � � scaling: -0.5 -0.25 and u = t ′ /t , P ( x, t ′ ) ≃ Ku 1 / γ P Ku 1 / γ , t P(x,t=100) 0e+00 -2 erence between 10 | x | = v 0 t ballistic peaks: -1 -0.5 0 1000 2000 0 1000 2000 0 1000 2000 i i i -4 FPU 10 0 50 100 150 x
Random walks: active media position of the w equation ˙ x = v 0 + ξ ( t ), Gaussian process of vanishing mean Gaussian process of vanishing i.e., � ξ ( t ) ξ ( s ) � = D v δ ( t − s ). wn biased Wiener process x ( t + τ ) = x ( t ) + v 0 τ + w ( τ ) , p ( w, τ ) Gaussian PDF � -1/2 -2 10 P hump ( x, t ) = Φ ( t ) [ p ( x + v 0 t, t ) + p ( x − v 0 t, t )] / 2 p(x,t)t 0 10 vior Φ ( t ) ∝ ( t/ τ 0 ) 1 − γ -4 10 � p(x,t) t the ballistic humps (4) x, t ′ ) ≃ u − 1 / 2 P hump (¯ x/u γ − 1 / 2 , t ) , and u = t ′ /t , -50 0 50 P hump (¯ -2 10 1/2 erence between x/t √ width of the hump ∝ t -4 10 0 100 200 300 � x/t
Hard point gas p : times t ¼ 1000 , 2000, 4000, and 6000 (the ffiffiffi v 0 ; D � / " -2 -2 10 10 ε = 4 % " ð x; t Þ ¼ % " 0 ð x; t=s 0 Þ ; -5 ε = 2 10 ε = 1 where s 0 ¼ p ffiffiffiffiffiffiffiffiffiffi " 0 =" (see ffiffiffiffiffiffiffiffiffiffi -6 10 -200 0 200 1/2 ) γ ρ (i,t) t is � ¼ 5 = 3 . -4 ρ (i,t/ ε -4 10 10 a) b) -6 -6 10 10 0 1000 0 2000 γ i i/t
FPU is � ¼ 5 = 3 . -2 -2 1 x 10 10 � � P ( x, t ′ ) ≃ Ku 1 / γ P Ku 1 / γ , t t = 1000 ε = 4 ε = 3 e(i,t) ε = 1 that K / � 1 � 1 = � -4 v 0 , 10 -3 10 0 a) when � 0 / v � = ð 1 � � Þ . e(i,t) 0 we demonstrated that t = 2000 -4 10 e(i,t) -4 10 b) c) -5 10 -6 10 0 1000 2000 3000 0 200 400 800 1000 600 i i Single particle PDF describes the energy correlation in many particle system
Conclusions Levy-walk-like dynamics in ergodic many particle systems New scalings - relation to phonon/mode coupling theories Velocity correlations can be calculated analytically Additional measurement/diagnostic tool [V. Zaburdaev, S. Denisov, and P. Hanggi PRL 106 180601 (2011)]
week ending P H Y S I C A L R E V I E W L E T T E R S V OLUME 91, N UMBER 19 7 NOVEMBER 2003 Dynamical Heat Channels S. Denisov, J. Klafter, and M. Urbakh School of Chemistry, Tel-Aviv University, Tel-Aviv 69978, Israel (Received 11 June 2003; published 4 November 2003) We consider heat conduction in a 1D dynamical channel. The channel consists of an ensemble of noninteracting particles, which move between two heat baths according to some dynamical process. We show that the essential thermodynamic properties of the heat channel can be obtained from the diffusion properties of the underlying particles. Emphasis is put on the conduction under anomalous diffusion conditions. DOI: 10.1103/PhysRevLett.91.194301 PACS numbers: 44.15.+a, 05.45.Ac, 05.60.Cd The link between thermodynamic phenomena and mi- the heat bath, the particle is ejected back to the channel croscopic dynamical chaos has been a subject of interest with a velocity v Th , which is chosen from P T � v Th � . for a long time [1]. An example of such a relationship is As particles do not interact, the dynamics of the en- deterministic diffusion [2], where local dynamical prop- semble inside the channel can be described by a long erties, such as stability (or instability) of several fixed trajectory of a single particle and the flux should be rescaled by the factor N . The trajectory of a particle is points can change the global diffusion from normal to independent of the particle velocity v Th . The only differ- anomalous [3]. Another question which has attracted a lot of attention is the problem of heat conductivity in deter- ence between ‘‘hot’’ and ‘‘cold’’ particles is that the hot ministic extended systems [4–8]. A large number of ones cover the same trajectories faster than the cold ones. The velocity v Th does not change during the propagation models have been proposed in order to understand the conditions under which a system obeys the Fourier heat through the channel and it can be interpreted as a tem- conduction law [4]. Recently, a new class of 1D models, perature ‘‘label’’ of a particle. The dependence of the dynamics in Eq. (1) on v Th can be taken into account by ‘‘billiard gas channels,’’ has been proposed [5–8]. These channels consist of two parallel walls with a series of introducing a scaling factor for the time t ! t=v Th . scatterers, distributed along walls, and noninteracting Because of the separation of the thermodynamic particles that move inside. The two ends of the channel characteristics from the dynamical ones, the proposed are in contact with heat baths. By changing the shapes approach is not limited to Hamiltonian systems only and positions of scatterers, it is possible to change the [5–8]. We assume only that the dynamics inside the conductivity of the channel [5–8]. channel has a diffusional character and can be charac- In this Letter we show that such billiard gas channels terized by the evolution of the mean square displace- belong to a wider class of models, which we call dynami- ment (msd) cal heat channels. The absence of interactions between h X 2 � t �i � t � : (2) particles and the independence of the particle dynamics This diffusion can be normal ( � � 1 ), subdiffusive on the kinetic energy allow a complete separation be- ( � < 1 ), or superdiffusive ( � > 1 ) [9]. tween the thermodynamic aspect, which is governed by Following Ref. [5], heat transfer by a particle through the properties of the thermostats, and the dynamics inside the channel is channel, which is governed by diffusion properties within the channel. All the essential information on heat con- M � t � M � t � X X q j � E in j � E out ductivity of such a dynamical heat conductor can be Q � t � � � E j � j � ; (3) obtained from the diffusion properties of the channel. j � 1 j � 1 Dynamical heat channel. —To model dynamics within where E in j and E out are the energies before and after the j j the channel, we consider N particles that move along collision with the heat bath. q j is the direction factor: direction X , following the equations of motion q j � 1 if the � j � 1 � collision is with the hot end, and q j � � 1 in the case of the cold end. M � t � is the total x � f � x ; t � ; _ X 2 x ; (1) number of collision events during time t . In the case of where the function f can be either deterministic or normal heat conductivity, Q grows linearly with t and the random. heat flux is defined by To consider transport of heat, two heat baths with Q � t � temperatures T � and T � are attached to the left and right J � lim : (4) ends of the channel. Each heat bath is characterized by a t t !1 velocity probability density function (pdf), P T � v Th � , Let us start from a situation where the particle is where v Th is the thermal velocity. After colliding with initially located at the hot end. During diffusion it can 0031-9007 = 03 = 91(19) = 194301(4)$20.00 194301-1 2003 The American Physical Society 194301-1
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