Statistical Physics of Synchronization Shamik Gupta Ramakrishna - PowerPoint PPT Presentation

Statistical Physics of Synchronization Shamik Gupta Ramakrishna Mission Vivekananda Educational and Research Institute, Kolkata, INDIA & ICTP, Trieste, ITALY (Regular Associate, Quantitative Life Sciences Section) Collaborators: ◮ Stefano Ruffo (Trieste) ◮ Alessandro Campa (Rome) ◮ Debraj Das (Kolkata)

The Kuramoto model 1. N globally coupled oscillators with distributed natural frequencies. i -th 2. θ i : Phase. oscillator � N � d θ i K θ i 3. dt = ω i + j =1 sin( θ j − θ i ). N 4. � K : Coupling constant, ω i ’s: Natural frequencies, Unimodal distr. g ( ω ).

The Kuramoto model 1. N globally coupled oscillators with distributed natural frequencies. i -th 2. θ i : Phase. oscillator � N � d θ i K θ i 3. dt = ω i + j =1 sin( θ j − θ i ). N 4. � K : Coupling constant, ω i ’s: Natural frequencies, Unimodal distr. g ( ω ). ◮ N → ∞ , t → ∞ limit: � N r 1. Define r = | 1 j =1 e i θ j | . N 2. High � 1 K : Synchronized phase, r � = 0. 3. Low � K : Incoherent phase, r = 0. 4. “Phase transition” (Bifurcation) on tuning � K . 5. � � 2 0 K K c = π g ( � ω � ) . � K c

Nonlinear Dynamics vis-` a-vis Statistical Physics approach Nonlinear dynamics: 1. Deterministic equations of motion d x d t = f ( x ). 2. Study, on a case-by-case basis, x as a function of t ; Interest: limit t → ∞ .

Nonlinear Dynamics vis-` a-vis Statistical Physics approach Nonlinear dynamics: Statistical Physics: 1. Deterministic 1. Stochastic equations of motion, equations of motion e.g., a Hamiltonian system d x d t = f ( x ). with no external drive + environment (heat bath at temp. T ): 2. Study, on a d q d t = p , case-by-case basis, d p d t = Force derived from Hamiltonian x as a function of t ; − γ p + η ( t ) . Interest: � �� � limit t → ∞ . Effect of environment Langevin: Gaussian white noise η ( t ): � η ( t ) � = 0 , � η ( t ) η ( t ′ ) � = 2 D δ ( t − t ′ ).

Nonlinear Dynamics vis-` a-vis Statistical Physics approach Nonlinear dynamics: Statistical Physics: 1. Deterministic 1. Stochastic equations of motion, equations of motion e.g., a Hamiltonian system d x d t = f ( x ). with no external drive + environment (heat bath at temp. T ): 2. Study, on a d q d t = p , case-by-case basis, d p d t = Force derived from Hamiltonian x as a function of t ; − γ p + η ( t ) . Interest: � �� � limit t → ∞ . Effect of environment Langevin: Gaussian white noise η ( t ): � η ( t ) � = 0 , � η ( t ) η ( t ′ ) � = 2 D δ ( t − t ′ ). 2. Study distribution P ( x , t ) ≡ P ( q , p , t ) as a function of t .

Nonlinear Dynamics vis-` a-vis Statistical Physics approach Nonlinear dynamics: Statistical Physics: 1. Deterministic 1. Stochastic equations of motion, equations of motion e.g., a Hamiltonian system d x d t = f ( x ). with no external drive + environment (heat bath at temp. T ): 2. Study, on a d q d t = p , case-by-case basis, d p d t = Force derived from Hamiltonian x as a function of t ; − γ p + η ( t ) . Interest: � �� � limit t → ∞ . Effect of environment Langevin: Gaussian white noise η ( t ): � η ( t ) � = 0 , � η ( t ) η ( t ′ ) � = 2 D δ ( t − t ′ ). 2. Study distribution P ( x , t ) ≡ P ( q , p , t ) as a function of t . 3. D = γ k B T (Fluc.-Dissipation Th.) ⇒ P ( x , t → ∞ ) ∝ exp( − H / T ) (no need to solve the dynamics).

Nonlinear Dynamics vis-` a-vis Statistical Physics approach Nonlinear dynamics: Statistical Physics: 1. Deterministic 1. Stochastic equations of motion, equations of motion e.g., a Hamiltonian system d x d t = f ( x ). with no external drive + environment (heat bath at temp. T ): 2. Study, on a d q d t = p , case-by-case basis, d p d t = Force derived from Hamiltonian x as a function of t ; Interest: − γ p + η ( t ) . � �� � limit t → ∞ . Effect of environment Langevin: Gaussian white noise η ( t ): � η ( t ) � = 0 , � η ( t ) η ( t ′ ) � = 2 D δ ( t − t ′ ). 2. D = γ k B T (Fluc.-Dissipation Th.) ⇒ P ( x , t → ∞ ) ∝ exp( − H / T ) (no need to solve the dynamics). 3. Useful concepts like free energy whose minimization yields equilibrium phases.

Our contributions from statphys perspective ◮ The key steps taken: 1. Including noise in the Kuramoto dynamics. 2. Interpreting the model as a long-range interacting system of particles with quenched disorder, driven out of equilibrium. 3. Employing tools of statistical mechanics and kinetic theory to study statics and dynamics.

Our contributions from statphys perspective ◮ The key steps taken: 1. Including noise in the Kuramoto dynamics. 2. Interpreting the model as a long-range interacting system of particles with quenched disorder, driven out of equilibrium. 3. Employing tools of statistical mechanics and kinetic theory to study statics and dynamics.

Our contributions from statphys perspective ◮ The key steps taken: 1. Including noise in the Kuramoto dynamics. 2. Interpreting the model as a long-range interacting system of particles with quenched disorder, driven out of equilibrium. 3. Employing tools of statistical mechanics and kinetic theory to study statics and dynamics. ◮ ...and the main results: 1. Proving that the system relaxes to a nonequilibrium steady state (NESS) at long times. 2. Developing an exact analytical approach to compute the steady state distr. 3. By considering generalized Kuramoto dynamics, demonstrating with exact results a very rich phase diagram with eqlbm. and noneqlbm. transitions.

Our setting: The generalized Kuramoto model 1. N globally coupled rotors.

Our setting: The generalized Kuramoto model 1. N globally coupled rotors.

Our setting: The generalized Kuramoto model 1. N globally coupled rotors. 2. 2 dynamical variables: i -th oscillator θ i : Phase, θ i v i : Angular velocity.

Our setting: The generalized Kuramoto model 1. N globally coupled rotors. 2. 2 dynamical variables: i -th oscillator θ i : Phase, θ i v i : Angular velocity. d θ i 3. dt = v i .

Our setting: The generalized Kuramoto model 1. N globally coupled rotors. 2. 2 dynamical variables: i -th oscillator θ i : Phase, θ i v i : Angular velocity. d θ i 3. dt = v i . N � dt = K 4. m dv i sin( θ j − θ i ) . N j =1 � �� � Long − range interaction

Our setting: The generalized Kuramoto model 1. N globally coupled rotors. 2. 2 dynamical variables: i -th oscillator θ i : Phase, θ i v i : Angular velocity. d θ i 3. dt = v i . N � dt = K 4. m dv i sin( θ j − θ i ) . N j =1 � �� � Long − range interaction ◮ m : Moment of inertia, K : Coupling constant.

Our setting: The generalized Kuramoto model 1. N globally coupled rotors. 2. 2 dynamical variables: i -th oscillator θ i : Phase, θ i v i : Angular velocity. d θ i 3. dt = v i . N � dt = K 4. m dv i sin( θ j − θ i ) . N j =1 � �� � Long − range interaction ◮ m : Moment of inertia, K : Coupling constant. ◮ Hamiltonian dynamics: H = � N � N p 2 2 m + K i , j =1 [1 − cos( θ i − θ j )], p i = mv i . i i =1 2 N

Our setting: The generalized Kuramoto model 1. N globally coupled rotors. 2. 2 dynamical variables: i -th θ i : Phase, oscillator v i : Angular velocity. θ i d θ i 3. dt = v i . 4. m dv i dt = N � + K − γ v i sin( θ j − θ i ) + η i ( t ) . � �� � N ���� j =1 Damping Noise � �� � Long − range interaction ◮ m : Moment of inertia, K : Coupling constant, γ : Friction constant. ◮ Hamiltonian + heat bath. ◮ Gaussian white noise: � η i ( t ) � = 0 , � η i ( t ) η j ( t ′ ) � = 2 γ T δ ij δ ( t − t ′ ) T : Temperature of the heat bath.

Our setting: The generalized Kuramoto model 1. N globally coupled rotors. 2. 2 dynamical variables: i -th θ i : Phase, oscillator v i : Angular velocity. θ i d θ i 3. dt = v i . 4. m dv i dt = N � + K − γ v i sin( θ j − θ i ) + γω i + η i ( t ) . � �� � N ���� ���� j =1 Damping Drive ( Quenched disorder ) � �� � Noise Long − range interaction ◮ m : Moment of inertia, γ : Friction constant, K : Coupling constant. ◮ ω i ’s: Quenched random variables from a common distr. g ( ω ). ◮ Gaussian white noise: � η i ( t ) � = 0 , � η i ( t ) η j ( t ′ ) � = 2 γ T δ ij δ ( t − t ′ ).

Long-range interactions: A one-slide summary 1 V ( r ) ∼ r α ; 0 ≤ α ≤ d . Examples: Gravitation, Coulomb interaction,... Main distinguishing feature: Non-additivity. E Total � = E I + E II . I II Consequences: ◮ Statics: Ensemble inequivalence. ◮ Dynamics: Slow relaxation over times diverging with the system size. Physics of Long-Range Interacting Systems, Campa, Dauxois, Fanelli, Ruffo (Oxford, 2014)

Our setting: The generalized Kuramoto model d θ i dt = v i . ◮ ◮ m dv i dt = N � + K − γ v i sin( θ j − θ i ) + γω i + η i ( t ) . � �� � ���� N ���� j =1 Damping Drive ( Quenched disorder ) Noise � �� � Long − range interaction 1. g ( ω ) unimodal, with mean � ω and width σ .