Microscopic and Macroscopic Transport
Microscopic and Macroscopic Transport Alden Astwood
October 4, 2012
Microscopic and Macroscopic Transport
Outline
Microscopics: Quantum Electronic Transport Motivation Kenkre, Biscarini, and Bustamante Formalism Other work
Macroscopics: Collective Motion of Animals Motivation Centering Model Alignment Model
Microscopic and Macroscopic Transport
Microscopics: Motivation
Part I Microscopics
Microscopic and Macroscopic Transport
Microscopics: Motivation
Classical Conductors Drude-Sommerfeld theory Ohm’s law: Conductance proportional to width
Conductance
Ohm's Law Conductor
Width
Microscopic and Macroscopic Transport
Microscopics: Motivation
Quantum Conductors Can fabricate small, high mobility transistors Quantum effects can become important Very different transport properties
Conductance
Ballistic Conductor
Width
B. J. van Wees et al. PRL 60:848 (1988)
Microscopic and Macroscopic Transport
Microscopics: Motivation
Partially Coherent Motion
What happens in between? How can the motion be modeled? Ohm's Law Conductor
???
Width
Conductance
Conductance
Ballistic Conductor
Width
Microscopic and Macroscopic Transport
Microscopics: STM Theory
Partially Coherent Motion Quantum motion: Liouville-von Neumann equation: ∂ρ −i = [H, ρ] h¯ ∂t Partially coherent motion: stochastic Liouville equation A simple SLE: let off-diagonal elements decay ∂ ρmn −i = [H, ρ]mn − α(1 − δmn )ρmn ∂t h¯ Convert SLE into generalized Master equation d Pm (t) = − ∑ dt n
Z t 0
Amn (t − t0 )Pn (t0 )dt0
Microscopic and Macroscopic Transport
Microscopics: STM Theory
GME for Transport Calculations (KBB1) Must connect to reservoirs for current to flow GME for open system: R d Pm (t) = − ∑n 0t Amn (t − t0 )Pn (t0 )dt0 +δmS RS (t) −δmD RD (t) dt
1 Kenkre
(1995)
et al. PRB 51:11074 (1995), Biscarini et al. PRB 51:11089
Microscopic and Macroscopic Transport
Microscopics: STM Theory
GME for Transport Calculations (KBB1) Must connect to reservoirs for current to flow GME for open system: R d Pm (t) = − ∑n 0t Amn (t − t0 )Pn (t0 )dt0 +δmS RS (t) −δmD RD (t) dt
Internal Dynamics
1 Kenkre
(1995)
et al. PRB 51:11074 (1995), Biscarini et al. PRB 51:11089
Microscopic and Macroscopic Transport
Microscopics: STM Theory
GME for Transport Calculations (KBB1) Must connect to reservoirs for current to flow GME for open system: R d Pm (t) = − ∑n 0t Amn (t − t0 )Pn (t0 )dt0 +δmS RS (t) −δmD RD (t) dt
Internal Dynamics Inflow from source to site S
1 Kenkre
(1995)
et al. PRB 51:11074 (1995), Biscarini et al. PRB 51:11089
Microscopic and Macroscopic Transport
Microscopics: STM Theory
GME for Transport Calculations (KBB1) Must connect to reservoirs for current to flow GME for open system: R d Pm (t) = − ∑n 0t Amn (t − t0 )Pn (t0 )dt0 +δmS RS (t) −δmD RD (t) dt
Internal Dynamics Inflow from source to site S Outflow into drain from site D
1 Kenkre
(1995)
et al. PRB 51:11074 (1995), Biscarini et al. PRB 51:11089
Microscopic and Macroscopic Transport
Microscopics: STM Theory
GME for Transport Calculations (KBB1) Must connect to reservoirs for current to flow GME for open system: R d Pm (t) = − ∑n 0t Amn (t − t0 )Pn (t0 )dt0 +δmS RS (t) −δmD RD (t) dt
Internal Dynamics Inflow from source to site S Outflow into drain from site D What should the rates look like? 1 Kenkre
(1995)
et al. PRB 51:11074 (1995), Biscarini et al. PRB 51:11089
Microscopic and Macroscopic Transport
Microscopics: STM Theory
Constructing the Rates
Source and drain try to bring conductor to equilibrium Simple form: Pth S − PS (t) τS th P − PD (t) RD (t) = − D τD RS (t) =
Pth S : probability to find carrier at S when in equilibrium with source
Microscopic and Macroscopic Transport
Microscopics: STM Theory
Finding the Current
Current = Charge times Probability current: IS (t) = qRS (t) Rates depend on probabilities: need to solve GME “Defect technique”: Solve driven system in terms of isolated system’s propagators Use Laplace transform final value theorem to find steady state current
Microscopic and Macroscopic Transport
Microscopics: STM Theory
Steady State Current
Original KBB result: # " th − Pth η ηDth − Pth 1 S S D R= − ˜ DD (0) − Π ˜ DS (0) + τD Π ˜ SS (0) − Π ˜ SD (0) + τS 2 Π ηmth : isolated system equilibrium population ˜ mn : Laplace transformed propagators Π KBB assume no transient charge buildup
Microscopic and Macroscopic Transport
Microscopics: STM Theory
Extension of KBB Old result " # th − Pth η ηDth − Pth 1 S S D R= − ˜ DD (0) − Π ˜ DS (0) + τD Π ˜ SS (0) − Π ˜ SD (0) + τS 2 Π Should not assume no transient charge buildup! New steady state result: R=
th th ηDth Pth S − ηS PD ˜ SS (0) − Π ˜ SD (0) + τS ] + η th [Π ˜ DD (0) − Π ˜ DS (0) + τD ] ηDth [Π S
New time-dependent expression also derived
Microscopic and Macroscopic Transport
Microscopics: Other Work
Microscopics: Other Work
Other formalisms: Landauer-B¨uttiker transmission theory Non-equilibrium Green’s functions Wigner functions
New scattering calculations for some simplified systems Transport in disordered lattices (Classical) Effective Medium: Replace disordered Master equation with ordered GME New exact calculations for finite systems: long range memory at short times
Microscopic and Macroscopic Transport
Macroscopics: Motivation
Part II Macroscopics
Microscopic and Macroscopic Transport
Collective Motion: Challenges
Many interacting particles Out of equilibrium Nondeterministic Dynamical phase transition
Macroscopics: Motivation
Microscopic and Macroscopic Transport
Approaches to Modeling Individual-Level Models Dynamical (continuous time) Discrete time models
Continuum Models
Macroscopics: Motivation
Microscopic and Macroscopic Transport
Approaches to Modeling Individual-Level Models Dynamical (continuous time) Discrete time models
Continuum Models
Macroscopics: Motivation
Microscopic and Macroscopic Transport
Macroscopics: Motivation
Approaches to Modeling Individual-Level Models Dynamical (continuous time) Discrete time models
Continuum Models Example, Mikhailov and Zanette PRE 60:4571 (1999): 2 x¨ m = −(˙xm − 1)˙xm − γ ∑(xm − xn ) + Γm (t) n
Microscopic and Macroscopic Transport
Approaches to Modeling Individual-Level Models Dynamical (continuous time) Discrete time models
Continuum Models
Macroscopics: Motivation
Microscopic and Macroscopic Transport
Macroscopics: Motivation
Approaches to Modeling Individual-Level Models Dynamical (continuous time) Discrete time models
Continuum Models Example, Vicsek et al. PRL 75:1226 (1995): “at each time step a given particle driven with a constant absolute velocity assumes the average direction of motion of the particles in its neighborhood of radius r with some random perturbation added”
Microscopic and Macroscopic Transport
Approaches to Modeling Individual-Level Models Dynamical (continuous time) Discrete time models
Continuum Models
Macroscopics: Motivation
Microscopic and Macroscopic Transport
Macroscopics: Motivation
Approaches to Modeling Individual-Level Models Dynamical (continuous time) Discrete time models
Continuum Models Example, Toner and Tu PRL 75:4326 (1995): ∂t ρ + ∇ · (vρ) =0 ∂t v + (v · ∇)v =αv − β |v|2 v − ∇P + DL ∇(∇ · v) + D1 ∇2 v + D2 (v · ∇)2 v + f
Microscopic and Macroscopic Transport
Macroscopics: Motivation
Bridging the Gap
Is it possible to go from an individual-level description d xm (t) = ... dt to a continuum description? ∂ ρ(x, t) = ... ∂t Difficult to do analytically1
1 Czir´ ok
et al. PRL 82:209 (1999)
Microscopic and Macroscopic Transport
Macroscopics: Motivation
Hybrid Multiscale Approach
1 2
3
Begin with individual-level model Postulate equations of motion for continuous coarse-grained quantities Fit unknown parameters with simulation
C. William Gear. Comm. Math. Sci. 1:715 (2003); M. Raghib et al. J. Theor. Biol. 264:893 (2010)
Microscopic and Macroscopic Transport
Macroscopics: Motivation
A Different Approach
1 2 3 4
Simplify the model until analytic calculations can be done Solve as much as possible Coarse-grain to find continuous quantities Work backwards to write continuum equations of motion
Microscopic and Macroscopic Transport
Macroscopics: Centering Model
Macroscopics: Centering Model
Microscopic and Macroscopic Transport
Macroscopics: Centering Model
Mikhailov and Zanette Model (1999)
Equations of motion: 2 − 1)˙xm −γ ∑(xm − xn ) + Γm (t) . x¨ m = −(˙xm n
Mikhailov and Zanette PRE 60:4571 (1999)
Microscopic and Macroscopic Transport
Macroscopics: Centering Model
Mikhailov and Zanette Model (1999)
Equations of motion: 2 − 1)˙xm −γ ∑(xm − xn ) + Γm (t) . x¨ m = −(˙xm n
Self-Propulsion
Mikhailov and Zanette PRE 60:4571 (1999)
Microscopic and Macroscopic Transport
Macroscopics: Centering Model
Mikhailov and Zanette Model (1999)
Equations of motion: 2 − 1)˙xm −γ ∑(xm − xn ) + Γm (t) . x¨ m = −(˙xm n
Self-Propulsion Interaction (Centering)
Mikhailov and Zanette PRE 60:4571 (1999)
Microscopic and Macroscopic Transport
Macroscopics: Centering Model
Mikhailov and Zanette Model (1999)
Equations of motion: 2 − 1)˙xm −γ ∑(xm − xn ) + Γm (t) . x¨ m = −(˙xm n
Self-Propulsion Interaction (Centering) Noise
Mikhailov and Zanette PRE 60:4571 (1999)
Microscopic and Macroscopic Transport
Macroscopics: Centering Model
Step 1: Simplify
Simpler equation of motion: x˙ m = −γ ∑(xm − xn ) + vm + Γm (t) . n
Microscopic and Macroscopic Transport
Macroscopics: Centering Model
Step 1: Simplify
Simpler equation of motion: x˙ m = −γ ∑(xm − xn ) + vm + Γm (t) . n
Strong damping
Microscopic and Macroscopic Transport
Macroscopics: Centering Model
Step 1: Simplify
Simpler equation of motion: x˙ m = −γ ∑(xm − xn ) + vm + Γm (t) . n
Strong damping Centering interaction
Microscopic and Macroscopic Transport
Macroscopics: Centering Model
Step 1: Simplify
Simpler equation of motion: x˙ m = −γ ∑(xm − xn ) + vm + Γm (t) . n
Strong damping Centering interaction Bias
Microscopic and Macroscopic Transport
Macroscopics: Centering Model
Step 1: Simplify
Simpler equation of motion: x˙ m = −γ ∑(xm − xn ) + vm + Γm (t) . n
Strong damping Centering interaction Bias Noise with strength D0
Microscopic and Macroscopic Transport
Macroscopics: Centering Model
Step 2: Solve
Construct Fokker-Planck equation: ∂ ∂W ∂W ∂ 2W =γ∑ ∑(xm − xn)W − ∑ vm ∂ xm + D0 ∑ ∂ x2 . ∂t m m ∂ xm n m m W(x1 , ..., xN , t): full probability distribution This Fokker-Planck equation can be solved exactly
Microscopic and Macroscopic Transport
Macroscopics: Centering Model
Green’s Function
1 N−1 1−e−2Nγt 4πD0 t 2πD0 Nγ
G0 ({u}, {u0 }, t) = r (
2 Nγ um − u0m e−Nγt − u¯ 0 (1 − e−Nγt ) ∑ −2Nγt 2D0 (1 − e ) m 1 Nγ 1 0 −Nγt 0 −Nγt − − um − um e − u¯ (1 − e ) 4D0 t 2D0 (1 − e−2Nγt ) N ∑ m · exp
−
with um ≡ xm − v¯ t − vmNγ−¯v
Microscopic and Macroscopic Transport
Macroscopics: Centering Model
Centering Model: Dynamics
Center of mass drifts at v¯ =
1 N
∑m vm mth particle displaced from CM on average by Spreading due to noise
vm −¯v Nγ
Microscopic and Macroscopic Transport
Macroscopics: Centering Model
Step 3: Coarse Graining One coarse-grained quantity: density Density per Particle ρ(x, t) Let ρ(x, t)dx be the probability to find any particle between x and x + dx at time t, divided by N How does ρ evolve? Can be obtained from full distribution via 1 ρ(x, t) = ∑ N m
Z
δ (x − xm )W(x1 , ..., xN , t)dN x
Microscopic and Macroscopic Transport
Macroscopics: Centering Model
Density cont’d
First consider case with no bias (each vm is zero) Take initial condition with all particles at x = 0 Density is a Gaussian with variance 1 1 − e−2Nγt t 2 σ = 2D0 + 1− N N 2Nγ σ 2 ∼ 2D0 t at short times, σ 2 ∼ 2D0 t/N at long times Effective diffusion constant changes from D0 to D0 /N
Microscopic and Macroscopic Transport
Macroscopics: Centering Model
Step 4: Equation of Motion
ρ obeys a time-dependent diffusion equation: ∂ρ ∂ 2ρ = D(t) 2 ∂t ∂x Diffusion coefficient is half the derivative of the variance D(t) = D0 e−2Nγt +
D0 (1 − e−2Nγt ) N
Microscopic and Macroscopic Transport
Macroscopics: Centering Model
Density with Bias
Suppose we have “informed” and “uninformed” individuals Informed: vm = v Uninformed: vm = 0
Informed individuals pull ahead Uninformed individuals get dragged along Density is a sum of two Gaussians
Microscopic and Macroscopic Transport
Macroscopics: Centering Model
Density with Bias Total Density, t=0.013333 5
Density
4
3
2
1
0 -1
0
1
2 x
3
4
Microscopic and Macroscopic Transport
Macroscopics: Centering Model
Step 4: Equation of Motion Can no longer write time-dependent diffusion equation Must generalize to ∂ ρ(x, t) = ∂t
Z ∞ −∞
K (x − x0 , t)ρ(x0 , t)dx0
or ∂ ρ(x, t) = ∂t
Z tZ ∞ 0
−∞
M (x − x0 , t − t0 )ρ(x0 , t0 )dx0 dt0
Equation of motion is not unique Necessarily have non-locality in space!
Microscopic and Macroscopic Transport
Macroscopics: Alignment Model
Macroscopics: Alignment Model
Microscopic and Macroscopic Transport
Macroscopics: Alignment Model
Vicsek et al. 1995 Model Rule: “at each time step a given particle driven with a constant absolute velocity assumes the average direction of motion of the particles in its neighborhood of radius r with some random perturbation added”
Vicsek et al. PRL 75:1226 (1995)
Microscopic and Macroscopic Transport
Macroscopics: Alignment Model
Vicsek et al. 1995 Model Rule: “at each time step a given particle driven with a constant absolute velocity assumes the average direction of motion of the particles in its neighborhood of radius r with some random perturbation added” Self-Propulsion
Vicsek et al. PRL 75:1226 (1995)
Microscopic and Macroscopic Transport
Macroscopics: Alignment Model
Vicsek et al. 1995 Model Rule: “at each time step a given particle driven with a constant absolute velocity assumes the average direction of motion of the particles in its neighborhood of radius r with some random perturbation added” Self-Propulsion Alignment Interaction
Vicsek et al. PRL 75:1226 (1995)
Microscopic and Macroscopic Transport
Macroscopics: Alignment Model
Vicsek et al. 1995 Model Rule: “at each time step a given particle driven with a constant absolute velocity assumes the average direction of motion of the particles in its neighborhood of radius r with some random perturbation added” Self-Propulsion Alignment Interaction Noise
Vicsek et al. PRL 75:1226 (1995)
Microscopic and Macroscopic Transport
Macroscopics: Alignment Model
Step 1: Simplify
Keep self propulsion, but restrict to 1D in space Birds can only point left or right
Make the interaction range infinite
Microscopic and Macroscopic Transport
Macroscopics: Alignment Model
Equations of Motion
Velocity = Speed times Direction x˙ m (t) = cσm (t) σm (t) is +1 or −1 The directions σm (t) evolve stochastically What are the equations of motion for the distribution function(s)?
Microscopic and Macroscopic Transport
Distribution Functions Need 2N functions Qσ (x1 , ..., xN , t)
Macroscopics: Alignment Model
Microscopic and Macroscopic Transport
Macroscopics: Alignment Model
Distribution Functions Need 2N functions Qσ (x1 , ..., xN , t) General form of evolution ∂ Qσ = −c ∑m σm ∂∂xm Qσ + ∑σ 0 wσ ,σ 0 Qσ 0 − wσ 0 ,σ Qσ ∂t
Microscopic and Macroscopic Transport
Macroscopics: Alignment Model
Distribution Functions Need 2N functions Qσ (x1 , ..., xN , t) General form of evolution ∂ Qσ = −c ∑m σm ∂∂xm Qσ + ∑σ 0 wσ ,σ 0 Qσ 0 − wσ 0 ,σ Qσ ∂t Drift
Microscopic and Macroscopic Transport
Macroscopics: Alignment Model
Distribution Functions Need 2N functions Qσ (x1 , ..., xN , t) General form of evolution ∂ Qσ = −c ∑m σm ∂∂xm Qσ + ∑σ 0 wσ ,σ 0 Qσ 0 − wσ 0 ,σ Qσ ∂t Drift Flipping wσ ,σ 0 : rate to go from σ 0 to σ
Microscopic and Macroscopic Transport
Macroscopics: Alignment Model
Distribution Functions Need 2N functions Qσ (x1 , ..., xN , t) General form of evolution ∂ Qσ = −c ∑m σm ∂∂xm Qσ + ∑σ 0 wσ ,σ 0 Qσ 0 − wσ 0 ,σ Qσ ∂t Drift Flipping wσ ,σ 0 : rate to go from σ 0 to σ Flipping rates: independent of position
Microscopic and Macroscopic Transport
Macroscopics: Alignment Model
Constructing Rates Flipping rates independent of position Treat σm s like Ising spins H(σ ) = −
∑ Jmnσmσn
m
October 4, 2012
Microscopic and Macroscopic Transport
Outline
Microscopics: Quantum Electronic Transport Motivation Kenkre, Biscarini, and Bustamante Formalism Other work
Macroscopics: Collective Motion of Animals Motivation Centering Model Alignment Model
Microscopic and Macroscopic Transport
Microscopics: Motivation
Part I Microscopics
Microscopic and Macroscopic Transport
Microscopics: Motivation
Classical Conductors Drude-Sommerfeld theory Ohm’s law: Conductance proportional to width
Conductance
Ohm's Law Conductor
Width
Microscopic and Macroscopic Transport
Microscopics: Motivation
Quantum Conductors Can fabricate small, high mobility transistors Quantum effects can become important Very different transport properties
Conductance
Ballistic Conductor
Width
B. J. van Wees et al. PRL 60:848 (1988)
Microscopic and Macroscopic Transport
Microscopics: Motivation
Partially Coherent Motion
What happens in between? How can the motion be modeled? Ohm's Law Conductor
???
Width
Conductance
Conductance
Ballistic Conductor
Width
Microscopic and Macroscopic Transport
Microscopics: STM Theory
Partially Coherent Motion Quantum motion: Liouville-von Neumann equation: ∂ρ −i = [H, ρ] h¯ ∂t Partially coherent motion: stochastic Liouville equation A simple SLE: let off-diagonal elements decay ∂ ρmn −i = [H, ρ]mn − α(1 − δmn )ρmn ∂t h¯ Convert SLE into generalized Master equation d Pm (t) = − ∑ dt n
Z t 0
Amn (t − t0 )Pn (t0 )dt0
Microscopic and Macroscopic Transport
Microscopics: STM Theory
GME for Transport Calculations (KBB1) Must connect to reservoirs for current to flow GME for open system: R d Pm (t) = − ∑n 0t Amn (t − t0 )Pn (t0 )dt0 +δmS RS (t) −δmD RD (t) dt
1 Kenkre
(1995)
et al. PRB 51:11074 (1995), Biscarini et al. PRB 51:11089
Microscopic and Macroscopic Transport
Microscopics: STM Theory
GME for Transport Calculations (KBB1) Must connect to reservoirs for current to flow GME for open system: R d Pm (t) = − ∑n 0t Amn (t − t0 )Pn (t0 )dt0 +δmS RS (t) −δmD RD (t) dt
Internal Dynamics
1 Kenkre
(1995)
et al. PRB 51:11074 (1995), Biscarini et al. PRB 51:11089
Microscopic and Macroscopic Transport
Microscopics: STM Theory
GME for Transport Calculations (KBB1) Must connect to reservoirs for current to flow GME for open system: R d Pm (t) = − ∑n 0t Amn (t − t0 )Pn (t0 )dt0 +δmS RS (t) −δmD RD (t) dt
Internal Dynamics Inflow from source to site S
1 Kenkre
(1995)
et al. PRB 51:11074 (1995), Biscarini et al. PRB 51:11089
Microscopic and Macroscopic Transport
Microscopics: STM Theory
GME for Transport Calculations (KBB1) Must connect to reservoirs for current to flow GME for open system: R d Pm (t) = − ∑n 0t Amn (t − t0 )Pn (t0 )dt0 +δmS RS (t) −δmD RD (t) dt
Internal Dynamics Inflow from source to site S Outflow into drain from site D
1 Kenkre
(1995)
et al. PRB 51:11074 (1995), Biscarini et al. PRB 51:11089
Microscopic and Macroscopic Transport
Microscopics: STM Theory
GME for Transport Calculations (KBB1) Must connect to reservoirs for current to flow GME for open system: R d Pm (t) = − ∑n 0t Amn (t − t0 )Pn (t0 )dt0 +δmS RS (t) −δmD RD (t) dt
Internal Dynamics Inflow from source to site S Outflow into drain from site D What should the rates look like? 1 Kenkre
(1995)
et al. PRB 51:11074 (1995), Biscarini et al. PRB 51:11089
Microscopic and Macroscopic Transport
Microscopics: STM Theory
Constructing the Rates
Source and drain try to bring conductor to equilibrium Simple form: Pth S − PS (t) τS th P − PD (t) RD (t) = − D τD RS (t) =
Pth S : probability to find carrier at S when in equilibrium with source
Microscopic and Macroscopic Transport
Microscopics: STM Theory
Finding the Current
Current = Charge times Probability current: IS (t) = qRS (t) Rates depend on probabilities: need to solve GME “Defect technique”: Solve driven system in terms of isolated system’s propagators Use Laplace transform final value theorem to find steady state current
Microscopic and Macroscopic Transport
Microscopics: STM Theory
Steady State Current
Original KBB result: # " th − Pth η ηDth − Pth 1 S S D R= − ˜ DD (0) − Π ˜ DS (0) + τD Π ˜ SS (0) − Π ˜ SD (0) + τS 2 Π ηmth : isolated system equilibrium population ˜ mn : Laplace transformed propagators Π KBB assume no transient charge buildup
Microscopic and Macroscopic Transport
Microscopics: STM Theory
Extension of KBB Old result " # th − Pth η ηDth − Pth 1 S S D R= − ˜ DD (0) − Π ˜ DS (0) + τD Π ˜ SS (0) − Π ˜ SD (0) + τS 2 Π Should not assume no transient charge buildup! New steady state result: R=
th th ηDth Pth S − ηS PD ˜ SS (0) − Π ˜ SD (0) + τS ] + η th [Π ˜ DD (0) − Π ˜ DS (0) + τD ] ηDth [Π S
New time-dependent expression also derived
Microscopic and Macroscopic Transport
Microscopics: Other Work
Microscopics: Other Work
Other formalisms: Landauer-B¨uttiker transmission theory Non-equilibrium Green’s functions Wigner functions
New scattering calculations for some simplified systems Transport in disordered lattices (Classical) Effective Medium: Replace disordered Master equation with ordered GME New exact calculations for finite systems: long range memory at short times
Microscopic and Macroscopic Transport
Macroscopics: Motivation
Part II Macroscopics
Microscopic and Macroscopic Transport
Collective Motion: Challenges
Many interacting particles Out of equilibrium Nondeterministic Dynamical phase transition
Macroscopics: Motivation
Microscopic and Macroscopic Transport
Approaches to Modeling Individual-Level Models Dynamical (continuous time) Discrete time models
Continuum Models
Macroscopics: Motivation
Microscopic and Macroscopic Transport
Approaches to Modeling Individual-Level Models Dynamical (continuous time) Discrete time models
Continuum Models
Macroscopics: Motivation
Microscopic and Macroscopic Transport
Macroscopics: Motivation
Approaches to Modeling Individual-Level Models Dynamical (continuous time) Discrete time models
Continuum Models Example, Mikhailov and Zanette PRE 60:4571 (1999): 2 x¨ m = −(˙xm − 1)˙xm − γ ∑(xm − xn ) + Γm (t) n
Microscopic and Macroscopic Transport
Approaches to Modeling Individual-Level Models Dynamical (continuous time) Discrete time models
Continuum Models
Macroscopics: Motivation
Microscopic and Macroscopic Transport
Macroscopics: Motivation
Approaches to Modeling Individual-Level Models Dynamical (continuous time) Discrete time models
Continuum Models Example, Vicsek et al. PRL 75:1226 (1995): “at each time step a given particle driven with a constant absolute velocity assumes the average direction of motion of the particles in its neighborhood of radius r with some random perturbation added”
Microscopic and Macroscopic Transport
Approaches to Modeling Individual-Level Models Dynamical (continuous time) Discrete time models
Continuum Models
Macroscopics: Motivation
Microscopic and Macroscopic Transport
Macroscopics: Motivation
Approaches to Modeling Individual-Level Models Dynamical (continuous time) Discrete time models
Continuum Models Example, Toner and Tu PRL 75:4326 (1995): ∂t ρ + ∇ · (vρ) =0 ∂t v + (v · ∇)v =αv − β |v|2 v − ∇P + DL ∇(∇ · v) + D1 ∇2 v + D2 (v · ∇)2 v + f
Microscopic and Macroscopic Transport
Macroscopics: Motivation
Bridging the Gap
Is it possible to go from an individual-level description d xm (t) = ... dt to a continuum description? ∂ ρ(x, t) = ... ∂t Difficult to do analytically1
1 Czir´ ok
et al. PRL 82:209 (1999)
Microscopic and Macroscopic Transport
Macroscopics: Motivation
Hybrid Multiscale Approach
1 2
3
Begin with individual-level model Postulate equations of motion for continuous coarse-grained quantities Fit unknown parameters with simulation
C. William Gear. Comm. Math. Sci. 1:715 (2003); M. Raghib et al. J. Theor. Biol. 264:893 (2010)
Microscopic and Macroscopic Transport
Macroscopics: Motivation
A Different Approach
1 2 3 4
Simplify the model until analytic calculations can be done Solve as much as possible Coarse-grain to find continuous quantities Work backwards to write continuum equations of motion
Microscopic and Macroscopic Transport
Macroscopics: Centering Model
Macroscopics: Centering Model
Microscopic and Macroscopic Transport
Macroscopics: Centering Model
Mikhailov and Zanette Model (1999)
Equations of motion: 2 − 1)˙xm −γ ∑(xm − xn ) + Γm (t) . x¨ m = −(˙xm n
Mikhailov and Zanette PRE 60:4571 (1999)
Microscopic and Macroscopic Transport
Macroscopics: Centering Model
Mikhailov and Zanette Model (1999)
Equations of motion: 2 − 1)˙xm −γ ∑(xm − xn ) + Γm (t) . x¨ m = −(˙xm n
Self-Propulsion
Mikhailov and Zanette PRE 60:4571 (1999)
Microscopic and Macroscopic Transport
Macroscopics: Centering Model
Mikhailov and Zanette Model (1999)
Equations of motion: 2 − 1)˙xm −γ ∑(xm − xn ) + Γm (t) . x¨ m = −(˙xm n
Self-Propulsion Interaction (Centering)
Mikhailov and Zanette PRE 60:4571 (1999)
Microscopic and Macroscopic Transport
Macroscopics: Centering Model
Mikhailov and Zanette Model (1999)
Equations of motion: 2 − 1)˙xm −γ ∑(xm − xn ) + Γm (t) . x¨ m = −(˙xm n
Self-Propulsion Interaction (Centering) Noise
Mikhailov and Zanette PRE 60:4571 (1999)
Microscopic and Macroscopic Transport
Macroscopics: Centering Model
Step 1: Simplify
Simpler equation of motion: x˙ m = −γ ∑(xm − xn ) + vm + Γm (t) . n
Microscopic and Macroscopic Transport
Macroscopics: Centering Model
Step 1: Simplify
Simpler equation of motion: x˙ m = −γ ∑(xm − xn ) + vm + Γm (t) . n
Strong damping
Microscopic and Macroscopic Transport
Macroscopics: Centering Model
Step 1: Simplify
Simpler equation of motion: x˙ m = −γ ∑(xm − xn ) + vm + Γm (t) . n
Strong damping Centering interaction
Microscopic and Macroscopic Transport
Macroscopics: Centering Model
Step 1: Simplify
Simpler equation of motion: x˙ m = −γ ∑(xm − xn ) + vm + Γm (t) . n
Strong damping Centering interaction Bias
Microscopic and Macroscopic Transport
Macroscopics: Centering Model
Step 1: Simplify
Simpler equation of motion: x˙ m = −γ ∑(xm − xn ) + vm + Γm (t) . n
Strong damping Centering interaction Bias Noise with strength D0
Microscopic and Macroscopic Transport
Macroscopics: Centering Model
Step 2: Solve
Construct Fokker-Planck equation: ∂ ∂W ∂W ∂ 2W =γ∑ ∑(xm − xn)W − ∑ vm ∂ xm + D0 ∑ ∂ x2 . ∂t m m ∂ xm n m m W(x1 , ..., xN , t): full probability distribution This Fokker-Planck equation can be solved exactly
Microscopic and Macroscopic Transport
Macroscopics: Centering Model
Green’s Function
1 N−1 1−e−2Nγt 4πD0 t 2πD0 Nγ
G0 ({u}, {u0 }, t) = r (
2 Nγ um − u0m e−Nγt − u¯ 0 (1 − e−Nγt ) ∑ −2Nγt 2D0 (1 − e ) m 1 Nγ 1 0 −Nγt 0 −Nγt − − um − um e − u¯ (1 − e ) 4D0 t 2D0 (1 − e−2Nγt ) N ∑ m · exp
−
with um ≡ xm − v¯ t − vmNγ−¯v
Microscopic and Macroscopic Transport
Macroscopics: Centering Model
Centering Model: Dynamics
Center of mass drifts at v¯ =
1 N
∑m vm mth particle displaced from CM on average by Spreading due to noise
vm −¯v Nγ
Microscopic and Macroscopic Transport
Macroscopics: Centering Model
Step 3: Coarse Graining One coarse-grained quantity: density Density per Particle ρ(x, t) Let ρ(x, t)dx be the probability to find any particle between x and x + dx at time t, divided by N How does ρ evolve? Can be obtained from full distribution via 1 ρ(x, t) = ∑ N m
Z
δ (x − xm )W(x1 , ..., xN , t)dN x
Microscopic and Macroscopic Transport
Macroscopics: Centering Model
Density cont’d
First consider case with no bias (each vm is zero) Take initial condition with all particles at x = 0 Density is a Gaussian with variance 1 1 − e−2Nγt t 2 σ = 2D0 + 1− N N 2Nγ σ 2 ∼ 2D0 t at short times, σ 2 ∼ 2D0 t/N at long times Effective diffusion constant changes from D0 to D0 /N
Microscopic and Macroscopic Transport
Macroscopics: Centering Model
Step 4: Equation of Motion
ρ obeys a time-dependent diffusion equation: ∂ρ ∂ 2ρ = D(t) 2 ∂t ∂x Diffusion coefficient is half the derivative of the variance D(t) = D0 e−2Nγt +
D0 (1 − e−2Nγt ) N
Microscopic and Macroscopic Transport
Macroscopics: Centering Model
Density with Bias
Suppose we have “informed” and “uninformed” individuals Informed: vm = v Uninformed: vm = 0
Informed individuals pull ahead Uninformed individuals get dragged along Density is a sum of two Gaussians
Microscopic and Macroscopic Transport
Macroscopics: Centering Model
Density with Bias Total Density, t=0.013333 5
Density
4
3
2
1
0 -1
0
1
2 x
3
4
Microscopic and Macroscopic Transport
Macroscopics: Centering Model
Step 4: Equation of Motion Can no longer write time-dependent diffusion equation Must generalize to ∂ ρ(x, t) = ∂t
Z ∞ −∞
K (x − x0 , t)ρ(x0 , t)dx0
or ∂ ρ(x, t) = ∂t
Z tZ ∞ 0
−∞
M (x − x0 , t − t0 )ρ(x0 , t0 )dx0 dt0
Equation of motion is not unique Necessarily have non-locality in space!
Microscopic and Macroscopic Transport
Macroscopics: Alignment Model
Macroscopics: Alignment Model
Microscopic and Macroscopic Transport
Macroscopics: Alignment Model
Vicsek et al. 1995 Model Rule: “at each time step a given particle driven with a constant absolute velocity assumes the average direction of motion of the particles in its neighborhood of radius r with some random perturbation added”
Vicsek et al. PRL 75:1226 (1995)
Microscopic and Macroscopic Transport
Macroscopics: Alignment Model
Vicsek et al. 1995 Model Rule: “at each time step a given particle driven with a constant absolute velocity assumes the average direction of motion of the particles in its neighborhood of radius r with some random perturbation added” Self-Propulsion
Vicsek et al. PRL 75:1226 (1995)
Microscopic and Macroscopic Transport
Macroscopics: Alignment Model
Vicsek et al. 1995 Model Rule: “at each time step a given particle driven with a constant absolute velocity assumes the average direction of motion of the particles in its neighborhood of radius r with some random perturbation added” Self-Propulsion Alignment Interaction
Vicsek et al. PRL 75:1226 (1995)
Microscopic and Macroscopic Transport
Macroscopics: Alignment Model
Vicsek et al. 1995 Model Rule: “at each time step a given particle driven with a constant absolute velocity assumes the average direction of motion of the particles in its neighborhood of radius r with some random perturbation added” Self-Propulsion Alignment Interaction Noise
Vicsek et al. PRL 75:1226 (1995)
Microscopic and Macroscopic Transport
Macroscopics: Alignment Model
Step 1: Simplify
Keep self propulsion, but restrict to 1D in space Birds can only point left or right
Make the interaction range infinite
Microscopic and Macroscopic Transport
Macroscopics: Alignment Model
Equations of Motion
Velocity = Speed times Direction x˙ m (t) = cσm (t) σm (t) is +1 or −1 The directions σm (t) evolve stochastically What are the equations of motion for the distribution function(s)?
Microscopic and Macroscopic Transport
Distribution Functions Need 2N functions Qσ (x1 , ..., xN , t)
Macroscopics: Alignment Model
Microscopic and Macroscopic Transport
Macroscopics: Alignment Model
Distribution Functions Need 2N functions Qσ (x1 , ..., xN , t) General form of evolution ∂ Qσ = −c ∑m σm ∂∂xm Qσ + ∑σ 0 wσ ,σ 0 Qσ 0 − wσ 0 ,σ Qσ ∂t
Microscopic and Macroscopic Transport
Macroscopics: Alignment Model
Distribution Functions Need 2N functions Qσ (x1 , ..., xN , t) General form of evolution ∂ Qσ = −c ∑m σm ∂∂xm Qσ + ∑σ 0 wσ ,σ 0 Qσ 0 − wσ 0 ,σ Qσ ∂t Drift
Microscopic and Macroscopic Transport
Macroscopics: Alignment Model
Distribution Functions Need 2N functions Qσ (x1 , ..., xN , t) General form of evolution ∂ Qσ = −c ∑m σm ∂∂xm Qσ + ∑σ 0 wσ ,σ 0 Qσ 0 − wσ 0 ,σ Qσ ∂t Drift Flipping wσ ,σ 0 : rate to go from σ 0 to σ
Microscopic and Macroscopic Transport
Macroscopics: Alignment Model
Distribution Functions Need 2N functions Qσ (x1 , ..., xN , t) General form of evolution ∂ Qσ = −c ∑m σm ∂∂xm Qσ + ∑σ 0 wσ ,σ 0 Qσ 0 − wσ 0 ,σ Qσ ∂t Drift Flipping wσ ,σ 0 : rate to go from σ 0 to σ Flipping rates: independent of position
Microscopic and Macroscopic Transport
Macroscopics: Alignment Model
Constructing Rates Flipping rates independent of position Treat σm s like Ising spins H(σ ) = −
∑ Jmnσmσn
m
