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Fundamental Physics from Control Theory? Reduced Order Models in Physics:

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The Really Big Picture Directions in Theoretical Physics (not exhaustive and highly subjective) small normal large

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Physical Focus of Talk Systems With Many Degrees of Freedom Natural bulk characteristics Theoretical techniques used to find bulk characteristics (physicists methods)

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Control Theory from an applied perspective

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Control Theory Focus of Talk Distributed Systems (high order systems, usually governed by PDEs) Model Reduction (related to finding approximate reduced order realizations) An issue of practical design

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Phenomena Physics The Quantum-Classical transition Molecular Dynamics (simulations) leading to STZ theory (understanding shearing in amorphous materials) Statistical Mechanics -- Thermodynamics Focus of Talk

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Stat Phys and Thermo Many Degrees of Freedom = Micro Statistical Description of System Natural Bulk Characteristics = Pressure, Volume, Temperature, Energy, … (i.e. thermodynamic quantities) Theoretical Techniques: Mean Field, Projection-operator methods, Renormalization Group (RG), …..

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An Example micro macro Tennis ball Monomers & Molecules Polymers (fibers)

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Reduction: The System- Environment Split Ingredients: Many state variables X=(x 1,x 2, …., x N ) Energy Conservation (i.e. for linear systems -- only has strictly imaginary eigenvalues) Insulating Walls (walls cannot act as an energy sink)

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Mathematical Caricature: Dynamics System-Environment split occurs when some state variables effectively decouple. (can result when there is an invariant subspace) i.e. X = (X s,X e ) System = Bulk Properties that are observed Environment = effectively is noise, the source of fluctuations in the system

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New Effective Dynamics: Approximate the environment contribution by a stochastic driving term, F(X S ) New Stochastic Dynamics: Result: A Langevin type equation (motivated by work by M. Kac and R. Zwanzig)

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RG: In a picture fine grained coarse grained Coarse Grained Variables = averaged variables

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RG: Heuristics System-Environment split in RG context System = Averaged variables Environment = The details that are ignored Example: Fine grained functional Coarse grained functional where

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Physics Reduction Comments: Pros: Quite generally applicable for closed systems A great calculational apparatus – may be applied to linear and nonlinear systems Quite algorithmic – easy to put on the computer Many implementations: Path integral RG, Density Matrix RG, Wilsonian RG, etc. Caveats: Open systems? Not often implemented for non-homogeneous systems Uncontrolled approximation Not very rigorous (at least in majority of literature)

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Noise gets translated into In physics: The coupling constants (in the Lagrangian) get RENORMALIZED Example: Charge Screening in electronic systems ASIDE: The above transformation may not be invertible (i.e. the RG transformations form a semi-group)

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A Control Theory Tutorial: Linear Systems X = The internal state of the system y = The output u = The input A = Determines the internal dynamics of the system B = Determines which states get externally excited C = Determines what quantities are measured

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Solution to such a system is: If X(0)=0, then 0 t The Names of G: Impulse Response Greens Function

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For Linear Time Invariant Causal Systems: Schematic form of the above equation Zero above diagonal Equal along the diagonals y =u T i = Toeplitz operator Γ = Hankel operator

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t -T Control from x(-T) = 0 to x(0) = x 0, with minimal input. NOTE: later C=Ψ c

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Quantifying Controllability Ψ c Ψ c * has the same range as Ψ c If the matrix Ψ c Ψ c * is invertible, then the system is controllable Small eigenvalues of Ψ c Ψ c * correspond to directions (states) that arent very controllable Singular values of Ψ c are related to the eigenvalues of Ψ c Ψ c * as so:

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t T Observe output. NOTE: later O=Ψ o

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Quantifying Observability Ψ o * Ψ o has the same null space as Ψ o If the matrix Ψ o * Ψ o is invertible, then the system is observable Small eigenvalues of Ψ o Ψ o correspond to directions (states) that arent very observable Singular values of Ψ o are related to the eigenvalues of Ψ o * Ψ o as so:

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t -T t T Simple input-output system Past inputs (t < 0) create state x(0) = x 0 at time t = 0. The input is shut off for t > 0. The output is observed for t > 0. Separating forcing from observing makes the math simple and accessible Key conclusions are relevant to more complicated situations

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t -T t T Hankel operators and singular values

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t -T t T

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Impulse response t -T t T Singular values: measure gain and approximate rank

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Intuition: H is a high-gain, low-rank operator (matrix). t -T t T

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t t T Optimal kth order model

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Model Reduction: Goal: Approximate the impulse response by a lower rank operator by using information from the Hankel operator (this scheme generalizes) Fact: When a system is controllable and observable, then one can find coordinates such that: Ψ o * Ψ o = Ψ c Ψ c * WHAT ADVANTAGE DO THESE COORDINATES GIVE US? and

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ANSWER: Controllability and observability are on the same footing The Hankel Singular Values may be directly interpreted in terms of oberservability and controllability RESULT: System = state variables that are very controllable and observable (i.e. correspond to large HSV) Environment = state variables that correspond to small HSV

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Example 1: The Heat Equation ln(σ n ) vs n More observable Less observable

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σ n vs n

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21N u=force y=velocity Mass=1 Spring K=1 Homogeneous N masses, N+1 springs (Work by Caltech group)

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Impulse response N=100 21N u=force y=velocity

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N=100 N=200 N=400

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states Full order

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states Full order

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Can get low order models with guaranteed error bounds. 6 states 40 states

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Semi-Summary Small HSVs correspond to environmental degrees of freedom Small HSVs are related to entropy and carry information about uncertainty and noise Small HSVs related to the observed dissipation Hints of fluctuation-dissipation theorem – without stochastic processes!

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Oddities: σ n vs n N = 20 springs in chain Approximate over reasonably short time scale C=B=I B=rank 1 C=I B=rank 1 C=rank 1

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σ n vs n N = 100 springs in chain Time scale on the order of the system length (mid scale) C=B=I B = rank 1 C=I

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σ n vs n N = 20 springs in chain Quite a long time scale (going like N 2 ) C=B=I C= rank 1 B= rank 1

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The End

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