10. Zwanzig-Mori Formalism
|
|
- Easter Gaines
- 5 years ago
- Views:
Transcription
1 University of Rhode Island Nonequilibrium Statistical Physics Physics Course Materials Zwanzig-Mori Formalism Gerhard Müller University of Rhode Island, Follow this and additional works at: nonequilibrium_statistical_physics Part of the Physics Commons Abstract Part ten of course materials for Nonequilibrium Statistical Physics (Physics 626), taught by Gerhard Müller at the University of Rhode Island. Entries listed in the table of contents, but not shown in the document, exist only in handwritten form. Documents will be updated periodically as more entries become presentable. Recommended Citation Müller, Gerhard, "0. Zwanzig-Mori Formalism" (205). Nonequilibrium Statistical Physics. Paper 0. This Course Material is brought to you for free and open access by the Physics Course Materials at It has been accepted for inclusion in Nonequilibrium Statistical Physics by an authorized administrator of For more information, please contact
2 Contents of this Document [ntc0] 0. Zwanzig-Mori Formalism Introduction [nln28] Time-dependence of expectation values (quantum and classical) Zwanzig s kinetic equation: generalized master equation [nln29] [nex68] Projection operator method (Mori formalism) [nln3] Kubo inner product [nln32] Projection operators [nln33] First and second projections [nln34] [nln35] Continued-fraction representation of relaxation function [nln36] Recursion method (algorithmic implementation of Mori formalism) Relaxation function with uniform continued-fraction coefficients [nex69] Continued-fraction expansion and moment expansion Generalized Langevin equation n-pole approximation Green s function formalism Structure function of harmonic oscillator [nex7], [nex72], [nex73] Scattering process and dynamic structure factor Electron scattering, neutron scattering, light scattering Scattering from a free atom Scattering from an atom bound in a harmonic potential Scattering from a harmonic crystal
3 Zwanzig-Mori formalism [nln28] Beginnings: Two phenomenological approaches for the dynamics of systems close to or at thermal equilibrium: Phenomenological equations of motion for probability distributions (e.g. master equation, Fokker-Planck equation). Phenomenological equations of motion for dynamical variables (e.g. Langevin equation) In these approaches, the focus is on selected degrees of freedom. All other degrees of freedom are taken into account summarily in the form of ad-hoc randomness. Completions: Microscopic foundations for these phenomenological approaches. Zwanzig (960): Rigorous derivation of a generalized master equation from first principles, i.e. from the Liouville equation. Mori (965): Rigorous derivation of a generalized Langevin equation from first principles, i.e. from the (quantum) Heisenberg equation or the (classical) canonical equations. The focus is again on selected degrees of freedom but here the effect of the other degrees of freedom are taken into account on a basis that is exact and amenable to systematic approximation. Variants: Zwanzig s approach leads to a kinetic equation of a particular kind. There exist alternative ways to derive kinetic equations from the Liouville equations via systematic approximations (e.g. via BBGKY hierarchy). Mori s approach has been formulated in more than one rendition. The version named projection operator formalism is most illuminating regarding the physical meaning of systematic approximations. The version named recursion method is most readily amenable to computational applications.
4 Zwanzig s method [nln29] Consider classical phase-space density ρ(t) = ρ(q, p ; q 2, p 2 ;... ; q n, p n ; t). Distinguish system (q, p ) and heat bath (q 2, p 2 ;... ; q n, p n ). Probability density of system: ρ (t) = ˆP ρ(t) via projection. Probability density of heat bath: ρ 2 (t) = ˆQρ(t), where ˆQ = ˆP. Implementation of projection: ρ = ˆP ρ(q, p ; q 2, p 2 ;... ; q n, p n ; t) = ρ eq (q 2, p 2 ;... ; q n, p n )σ(q, p ; t), where ρ eq / t = ıˆlρ eq = 0 and σ(q, p ; t) = dq 2 dp 2 dq n dp n ρ(q, p ; q 2, p 2 ;... ; q n, p n ; t). Idempotency, ˆP ρ (t) = ρ (t), satisfied by construction. Liouville equation, ρ(t)/ t = ıˆlρ(t), split into two coupled equations: ˆP t ρ(t) = t ρ (t) = ı ˆP ˆL [ ρ (t) + ρ 2 (t) ], () ˆQ t ρ(t) = t ρ 2(t) = ı ˆQˆL [ ρ (t) + ρ 2 (t) ]. (2) Formal solution of (2) [nex68] to be substituted into (): ρ 2 (t) = e ı ˆQˆLt ρ 2 (0) ı t 0 dτ e ı ˆQˆLτ ˆQˆLρ (t τ). Zwanzig s kinetic equation (generalized master equation): t ρ (t) = ı ˆP ˆLρ (t) ı ˆP ˆLe ı ˆQˆLt ρ 2 (0) t # Autonomous part of system s time evolution. 0 dτ ˆP ˆLe ı ˆQˆLτ ˆQˆLρ (t τ). #2 Autonomous part of heat bath s time evolution. Instantaneous effect of heat bath on system at time t. #3 Effect caused by system on heat bath at time t τ propagates in heat bath and feeds back into system at time t. First term often vanishes and second term (inhomogeneity) can be made zero by judicious choice of initial conditions. Zwanzig s kinetic equation can be used as the starting point for the derivation (via approximations) of a master equation or a Fokker-Planck equation.
5 [nex68] Zwanzig s kinetic equation In the derivation of Zwanzig s kinetic equation, t t ρ (t) = i ˆP ˆLρ (t) i ˆP ˆLe i ˆQ ˆLt ρ 2 (0) dτ ˆP ˆLe i ˆQ ˆLτ ˆQˆLρ (t τ), 0 from two projections of the Liouville equation, ˆP ρ t = ρ t = i ˆP ˆL[ρ + ρ 2 ], we use the formal solution ˆQ ρ t = ρ 2 t = i ˆQˆL[ρ + ρ 2 ], () Verify that (2) is a solution of (). Solution: t ρ 2 (t) = e i ˆQ ˆLt ρ 2 (0) i dτ e i ˆQ ˆLτ ˆQˆLρ (t τ). (2) 0
6 Projection operator method [nln3] Goal: Determination of symmetrized time-correlation function (fluctuation function) for a dynamical variable A(t) of a quantum or classical many-body Hamiltonian system H in thermal equilibrium. Fluctuation function (real, symmetric, normalized): C 0 (t). = A(t) A A A = A A( t) A A = A e Lt A. A A Dirac notation symbolizes inner product of choice as explained in [nln32]. Some properties of dynamic quantities depend on choice of inner product. Relaxation function (via Laplace transform): c 0 (z) = 0 dt e zt A e Lt A A A = A A A z + ıl A. Projection operator method determines relaxation function via systematic approximation. Inverse Laplace transform, C 0 (t) = 2πı C dz e zt c 0 (z), involves integral along straight path from ɛ ı to ɛ + ı for ɛ > 0. In practical applications, the (real, symmetric) spectral density is inferred from the relaxation function as limit process, Φ 0 (ω) = 2 lim ɛ 0 R{c 0 (ɛ ıω)}, and the fluctuation function via inverse Fourier transform, C 0 (t) = + dω 2π e ıωt Φ 0 (ω). The last bracket is also known as a Green s function.
7 Kubo inner product [nln32] General properties of inner products: A B = B A, A A = A 2 0, A λb = λ A B, A B + C = A B + A C. Kubo inner product for quantum system: where A B. = β β 0 dλ e λh A e λh B, A = Z Tr{e βh A}, Z = Tr{e βh }, β = k B T. Alternative inner product for quantum systems: A B. = 2 A B + BA. Both inner products have the same classical limit: A B =. d n q d n p e βh(q,p) A(q, p)b(q, p). Z Inner products of [nln3] employ... quantum Liouville operator, L = [L, ],, for time evolution governed by Heisenberg equation of motion, da dt = ı [H, A] = ıla. n ( H classical Liouville operator, L = ı{h, } = ı H q j= j p j p j q j for time evolution governed by Hamilton s equation of motion, da = {H, A} = ıla dt It is optional to subtract the product of expectation values A B in all inner products presented here ),
8 Projection operators [nln33] The relaxation function c 0 (z) is determined recursively by a succession of subdivisions of the many-body dynamics into components that are treated rigorously and a remainder that is treated phenomenologically. It is expected that the remainder diminishes in importance as the number of rigorous components is increased systematically. The time evolution of the dynamical variable A(t) can be conceived as a pirouette performed by the vector A(t) through the Hilbert space. The subdivisions are implemented by a sequence of projections onto onedimensional Hilbert subspaces traversed by A(t). Initial condition: f 0. = A(0) = A. Projection operators P n and Q n = P n, n = 0,,...,.. P 0 = f0 f 0 f 0 f 0, P0 2 = P 0, P 0 = P 0, P 0 Q 0 = Q 0 P 0 = 0. Orthogonal direction: f = ıl f 0, f 0 f = 0, P 0 f = 0, Q 0 f = f P 0 f = f, P = f f f f, Q = P. The systematic generation of further orthogonal direction will be discussed in the context of the recursion method. Successive projections filter out particular aspects of the many-body dynamics. The filters are applied in series. What passes through n filters is the remainder to be treated phenomenologically. The physical content of this process can be gleaned from the first two projections carried out in detail: First projection [nln33], Second projection [nln34]. Unitary transformation e ılt makes ıla orthogonal to A, implying A ıla = 0.
9 First projection [nln34] Rewrite relaxation function from [nln3] with projection operators from [nln33] and apply Dyson identity /(X + Y ) = /X (/X)Y [/(X + Y )]: f 0 f 0 c 0 (z) = f 0 z + ıl f 0 = f 0 z + ılp 0 + ılq 0 f 0 = f 0 z + ılq 0 f 0 + f 0 ılp 0 z + ılq 0 z + ıl f 0. Simplify both terms: [ f 0 + ( ı) z z LQ 0 + ( ı)2 f 0 ıl z + ılq 0 f 0 f 0 f 0 c 0 (z) = [ z f 0 ıl z + ılq 0 f 0 = f 0 = f 0 ( ı)lq 0 = ] LQ z 2 0 LQ 0 + f 0 f 0 z + ıl f 0 z + z, f 0 ıl f 0 f 0 z + ılq 0 f 0 = z f 0 f 0. = f 0 ıl z + ılq 0 f 0 c 0 (z). ] z (z + ılq 0 ) + ıl z + ılq 0 z + ılq 0 f 0 ıl z + ılq 0 f 0 = f 0 ( ı)lq 0 Q 0 ıl z + ıq 0 LQ 0 f 0 z + ıl f ; f = Q 0 f = Q 0 ıl f 0, L = Q 0 LQ 0. f Relaxation function after first projection expressed via memory function: c 0 (z) = z + Σ (z), Σ (z) = f f 0 f 0 z + ıl f. Memory function Σ (z) of original problem, {L, f 0 }, can be reinterpreted as the (as yet non-normalized) relaxation function of a new dynamical problem, {L, f }. Projection operator Q 0 acts as filter on the Liouvillian L, absorbing that part of dynamics dealt with explicitly in first projection. Explicit information contained in normalization constant of Σ (z). Direct consequence of operator identity (X + Y )/(X + Y ) =.
10 Second projection [nln35] Rewrite memory function from [nln34] with projection operators from [nln33] and apply Dyson identity: f 0 f 0 Σ (z) = f z + ıl f = f z + ıl P + ıl Q f = f z + ıl Q f f ıl P z + ıl Q z + ıl f = z f f f ıl z + ıl Q f f0 f 0 f f Σ (z), where simplifications analogous to [nln34] are carried out. Σ (z) = f f / f 0 f 0 z + z, f ıl f f z + ıl Q f z f ıl z + ıl Q f = = f ( ı)l Q Q ıl z + ıq L Q f = f 2 z + ıl 2 f 2, with f 2 = Q ıl f, L 2 = Q L Q. Memory function (first termination function) after second projection expressed via second termination function: Σ (z) = z + Σ 2 (z), Σ 2(z) = f 2 f f z + ıl 2 f 2 with = f f / f 0 f 0. The n th projection yields Σ n (z) = n z + Σ n (z), Σ n(z) = f n f n f n with n = f n f n / f n 2 f n 2 and f n = Q n ıl n f n, L n = Q n L n Q n. z + ıl n f n
11 Continued-fraction representation [nln36] Relaxation function after n successive projections: c 0 (z) = z + z + z n z + Σ n (z) The explicit dynamical information extracted from the original many-body system, {L, f 0 }, in the first n projections is contained in the continuedfraction coefficients:,..., n. Each projection adds a layer of projection operators around the original Liouvillian: L n = Q n Q 0 LQ 0 Q n. Expectation (somewhat naively): If n is sufficiently large, all distinctive spectral features of L will have been filtered out and incorporated explicitly into the relaxation function via the i. Whatever features of L still shine through the n filters are adequately represented by a source of white noise. The memory function associated with white noise is a constant, commonly represented by a relaxation time: Σ n (z) = τ n = onst. This completion of the continued fraction is known under the name n-pole approximation. The relaxation function is characterized by n poles in the complex frequency plane. Options used in practical applications: A number of continued-fraction coefficients are determined on phenomenological grounds along with a terminating relaxation time τ n. Examples: classical relaxator ( pole), classical oscillator (two poles). A number of continued-fraction coefficients are calculated from the original many-body system via the recursion method along with a termination function Σ n (z) inferred from an extrapolation scheme.
12 [nex69] Relaxation function with uniform continued-fraction coefficients. Find closed-form expressions for the relaxation function c 0 (z), the spectral density Φ 0 (ω), and the fluctuation function C 0 (t), of some physical system if we know that the (infinite) sequence of continued-fraction coefficients is (a) uniform: = 2 =... = 4 ω2 0, (b) almost uniform: = 2 ω2 0, 2 = 3... = 4 ω2 0. Solution:
13 [nex7] Structure function of harmonic oscillator I. Consider the quantum harmonic oscillator (for = ), H = p2 2m + 2 mω2 0q 2 = ω 0 ( a a + ), 2 where q = (a + a)/ 2mω 0, p = i mω 0 /2(a a) relate the position and momentum operators ([q, p] = i) to the boson creation and annihilation operators ([a, a ] = ). Use the recursion method with inner product (A, B) = 2 ( AB + BA ) to calculate the structure function S qq (ω) for the position variable at temperature T, where a a = n B = (e βω + ), β = /k B T. Solution:
14 [nex72] Structure function of harmonic oscillator II. Consider the classical harmonic oscillator, H = p2 2m + 2 mω2 0q 2. Use the recursion method with inner product (A, B) = AB to calculate the structure function S qq (ω) for the position variable at temperature T, where p 2 /2m = 2 mω2 0 q 2 = 2 k BT, according to the equipartition theorem. Solution:
15 [nex73] Structure function of harmonic oscillator III. Consider the quantum harmonic oscillator (for = ), H = p2 2m + 2 mω2 0q 2 = ω 0 ( a a + ), 2 where q = (a + a)/ 2mω 0, p = i mω 0 /2(a a) relate the position and momentum operators ([q, p] = i) to the boson creation and annihilation operators ([a, a ] = ). Solve the equation of motion ζ A; B ± ζ = [A, B] ± + [A, H]; B ± ζ for (a) the Green s function q; q ζ and (b) the Green s function q; q + ζ. Infer from each Green s function the structure function S qq (ω) for the position variable at temperature T. Solution:
10. Zwanzig-Mori Formalism
University of Rhode Island DigitalCommons@URI Nonequilibrium Statistical Physics Physics Course Materials 205 0. Zwanzig-Mori Formalism Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative
More information11. Recursion Method: Concepts
University of Rhode Island DigitalCommons@URI Nonequilibrium Statistical Physics Physics Course Materials 16 11. Recursion Method: Concepts Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative
More information08. Brownian Motion. University of Rhode Island. Gerhard Müller University of Rhode Island,
University of Rhode Island DigitalCommons@URI Nonequilibrium Statistical Physics Physics Course Materials 1-19-215 8. Brownian Motion Gerhard Müller University of Rhode Island, gmuller@uri.edu Follow this
More information09. Linear Response and Equilibrium Dynamics
University of Rhode Island DigitalCommons@URI Nonequilibrium Statistical Physics Physics Course Materials 2015 09. Linear Response and Equilibrium Dynamics Gerhard Müller University of Rhode Island, gmuller@uri.edu
More information04. Random Variables: Concepts
University of Rhode Island DigitalCommons@URI Nonequilibrium Statistical Physics Physics Course Materials 215 4. Random Variables: Concepts Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative
More information15. Hamiltonian Mechanics
University of Rhode Island DigitalCommons@URI Classical Dynamics Physics Course Materials 2015 15. Hamiltonian Mechanics Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative Commons License
More informationTable of Contents [ntc]
Table of Contents [ntc] 1. Introduction: Contents and Maps Table of contents [ntc] Equilibrium thermodynamics overview [nln6] Thermal equilibrium and nonequilibrium [nln1] Levels of description in statistical
More information5.74 Introductory Quantum Mechanics II
MIT OpenCourseWare http://ocw.mit.edu 5.74 Introductory Quantum Mechanics II Spring 9 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Andrei Tokmakoff,
More informationThis work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 4.0 License.
University of Rhode Island DigitalCommons@URI Classical Dynamics Physics Course Materials 015 14. Oscillations Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative Commons License This wor
More information14. Ideal Quantum Gases II: Fermions
University of Rhode Island DigitalCommons@URI Equilibrium Statistical Physics Physics Course Materials 25 4. Ideal Quantum Gases II: Fermions Gerhard Müller University of Rhode Island, gmuller@uri.edu
More information03. Simple Dynamical Systems
University of Rhode Island DigitalCommons@URI Classical Dynamics Physics Course Materials 2015 03. Simple Dynamical Systems Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative Commons License
More informationThis work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 4.0 License.
University of Rhode Island DigitalCommons@URI PHY 204: Elementary Physics II Physics Course Materials 2015 11. RC Circuits Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative Commons License
More informationDerivation of the GENERIC form of nonequilibrium thermodynamics from a statistical optimization principle
Derivation of the GENERIC form of nonequilibrium thermodynamics from a statistical optimization principle Bruce Turkington Univ. of Massachusetts Amherst An optimization principle for deriving nonequilibrium
More information13. Ideal Quantum Gases I: Bosons
University of Rhode Island DigitalCommons@URI Equilibrium Statistical Physics Physics Course Materials 5 3. Ideal Quantum Gases I: Bosons Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative
More information06. Lagrangian Mechanics II
University of Rhode Island DigitalCommons@URI Classical Dynamics Physics Course Materials 2015 06. Lagrangian Mechanics II Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative Commons License
More information10. Scattering from Central Force Potential
University of Rhode Island DigitalCommons@URI Classical Dynamics Physics Course Materials 215 1. Scattering from Central Force Potential Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative
More information12. Rigid Body Dynamics I
University of Rhode Island DigitalCommons@URI Classical Dynamics Physics Course Materials 015 1. Rigid Body Dynamics I Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative Commons License
More informationThis work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 4.0 License.
University of Rhode Island DigitalCommons@URI PHY 204: Elementary Physics II Physics Course Materials 2015 10. Resistors II Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative Commons License
More information02. Newtonian Gravitation
University of Rhode Island DigitalCommons@URI Classical Dynamics Physics Course Materials 2015 02. Newtonian Gravitation Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative Commons License
More informationA path integral approach to the Langevin equation
A path integral approach to the Langevin equation - Ashok Das Reference: A path integral approach to the Langevin equation, A. Das, S. Panda and J. R. L. Santos, arxiv:1411.0256 (to be published in Int.
More informationThis work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 4.0 License.
University of Rhode Island DigitalCommons@URI PHY 204: Elementary Physics II Physics Course Materials 2015 09. Resistors I Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative Commons License
More informationSpin-Boson Model. A simple Open Quantum System. M. Miller F. Tschirsich. Quantum Mechanics on Macroscopic Scales Theory of Condensed Matter July 2012
Spin-Boson Model A simple Open Quantum System M. Miller F. Tschirsich Quantum Mechanics on Macroscopic Scales Theory of Condensed Matter July 2012 Outline 1 Bloch-Equations 2 Classical Dissipations 3 Spin-Boson
More informationF(t) equilibrium under H 0
Physics 17b: Statistical Mechanics Linear Response Theory Useful references are Callen and Greene [1], and Chandler [], chapter 16. Task To calculate the change in a measurement B t) due to the application
More information13. Rigid Body Dynamics II
University of Rhode Island DigitalCommons@URI Classical Dynamics Physics Course Materials 2015 13. Rigid Body Dynamics II Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative Commons License
More informationFinite Temperature Field Theory
Finite Temperature Field Theory Dietrich Bödeker, Universität Bielefeld 1. Thermodynamics (better: thermo-statics) (a) Imaginary time formalism (b) free energy: scalar particles, resummation i. pedestrian
More information1 Equal-time and Time-ordered Green Functions
1 Equal-time and Time-ordered Green Functions Predictions for observables in quantum field theories are made by computing expectation values of products of field operators, which are called Green functions
More informationBest-fit quasi-equilibrium ensembles: a general approach to statistical closure of underresolved Hamiltonian dynamics
Best-fit quasi-equilibrium ensembles: a general approach to statistical closure of underresolved Hamiltonian dynamics Bruce Turkington University of Massachusetts Amherst and Petr Plecháč University of
More informationQuantum Mechanics I Physics 5701
Quantum Mechanics I Physics 5701 Z. E. Meziani 02/24//2017 Physics 5701 Lecture Commutation of Observables and First Consequences of the Postulates Outline 1 Commutation Relations 2 Uncertainty Relations
More informationLINEAR RESPONSE THEORY
MIT Department of Chemistry 5.74, Spring 5: Introductory Quantum Mechanics II Instructor: Professor Andrei Tokmakoff p. 8 LINEAR RESPONSE THEORY We have statistically described the time-dependent behavior
More informationMixing Quantum and Classical Mechanics: A Partially Miscible Solution
Mixing Quantum and Classical Mechanics: A Partially Miscible Solution R. Kapral S. Nielsen A. Sergi D. Mac Kernan G. Ciccotti quantum dynamics in a classical condensed phase environment how to simulate
More informationBrownian Motion and Langevin Equations
1 Brownian Motion and Langevin Equations 1.1 Langevin Equation and the Fluctuation- Dissipation Theorem The theory of Brownian motion is perhaps the simplest approximate way to treat the dynamics of nonequilibrium
More information06. Stochastic Processes: Concepts
University of Rhode Island DigitalCommons@URI Nonequilibrium Statistical Physics Physics Course Materials 2015 06. Stochastic Processes: Concepts Gerhard Müller University of Rhode Island, gmuller@uri.edu
More information9.1 System in contact with a heat reservoir
Chapter 9 Canonical ensemble 9. System in contact with a heat reservoir We consider a small system A characterized by E, V and N in thermal interaction with a heat reservoir A 2 characterized by E 2, V
More information4. The Green Kubo Relations
4. The Green Kubo Relations 4.1 The Langevin Equation In 1828 the botanist Robert Brown observed the motion of pollen grains suspended in a fluid. Although the system was allowed to come to equilibrium,
More informationThis work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 4.0 License.
University of Rhode Island DigitalCommons@URI PHY 204: Elementary Physics II Physics Course Materials 2015 18. RL Circuits Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative Commons License
More informationThis work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 4.0 License.
University of Rhode Island DigitalCommons@URI PHY 204: Elementary Physics II Physics Course Materials 2015 07. Capacitors I Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative Commons License
More informationPhysics 5153 Classical Mechanics. Canonical Transformations-1
1 Introduction Physics 5153 Classical Mechanics Canonical Transformations The choice of generalized coordinates used to describe a physical system is completely arbitrary, but the Lagrangian is invariant
More informationDISSIPATIVE TRANSPORT APERIODIC SOLIDS KUBO S FORMULA. and. Jean BELLISSARD 1 2. Collaborations:
Vienna February 1st 2005 1 DISSIPATIVE TRANSPORT and KUBO S FORMULA in APERIODIC SOLIDS Jean BELLISSARD 1 2 Georgia Institute of Technology, Atlanta, & Institut Universitaire de France Collaborations:
More information2.1 Green Functions in Quantum Mechanics
Chapter 2 Green Functions and Observables 2.1 Green Functions in Quantum Mechanics We will be interested in studying the properties of the ground state of a quantum mechanical many particle system. We
More information3.3 Energy absorption and the Green function
142 3. LINEAR RESPONSE THEORY 3.3 Energy absorption and the Green function In this section, we first present a calculation of the energy transferred to the system by the external perturbation H 1 = Âf(t)
More informationUnder evolution for a small time δt the area A(t) = q p evolves into an area
Physics 106a, Caltech 6 November, 2018 Lecture 11: Hamiltonian Mechanics II Towards statistical mechanics Phase space volumes are conserved by Hamiltonian dynamics We can use many nearby initial conditions
More informationSecond quantization: where quantization and particles come from?
110 Phys460.nb 7 Second quantization: where quantization and particles come from? 7.1. Lagrangian mechanics and canonical quantization Q: How do we quantize a general system? 7.1.1.Lagrangian Lagrangian
More informationThis work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 4.0 License.
University of Rhode Island DigitalCommons@URI Equilibrium Statistical Physics Physics Course Materials 2015 07. Kinetic Theory I Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative Commons
More information424 Index. Eigenvalue in quantum mechanics, 174 eigenvector in quantum mechanics, 174 Einstein equation, 334, 342, 393
Index After-effect function, 368, 369 anthropic principle, 232 assumptions nature of, 242 autocorrelation function, 292 average, 18 definition of, 17 ensemble, see ensemble average ideal,23 operational,
More informationNon-relativistic Quantum Electrodynamics
Rigorous Aspects of Relaxation to the Ground State Institut für Analysis, Dynamik und Modellierung October 25, 2010 Overview 1 Definition of the model Second quantization Non-relativistic QED 2 Existence
More informationSupplementary Information for Experimental signature of programmable quantum annealing
Supplementary Information for Experimental signature of programmable quantum annealing Supplementary Figures 8 7 6 Ti/exp(step T =.5 Ti/step Temperature 5 4 3 2 1 0 0 100 200 300 400 500 Annealing step
More information07. Stochastic Processes: Applications
University of Rhode Island DigitalCommons@URI Nonequilibrium Statistical Physics Physics Course Materials 10-19-2015 07. Stochastic Processes: Applications Gerhard Müller University of Rhode Island, gmuller@uri.edu
More information11 Perturbation Theory
S.K. Saikin Oct. 8, 009 11 Perturbation Theory Content: Variational Principle. Time-Dependent Perturbation Theory. 11.1 Variational Principle Lecture 11 If we need to compute the ground state energy of
More informationLecture 1: The Equilibrium Green Function Method
Lecture 1: The Equilibrium Green Function Method Mark Jarrell April 27, 2011 Contents 1 Why Green functions? 2 2 Different types of Green functions 4 2.1 Retarded, advanced, time ordered and Matsubara
More information4.3 Lecture 18: Quantum Mechanics
CHAPTER 4. QUANTUM SYSTEMS 73 4.3 Lecture 18: Quantum Mechanics 4.3.1 Basics Now that we have mathematical tools of linear algebra we are ready to develop a framework of quantum mechanics. The framework
More informationOn solving many-body Lindblad equation and quantum phase transition far from equilibrium
On solving many-body Lindblad equation and quantum phase transition far from equilibrium Department of Physics, FMF, University of Ljubljana, SLOVENIA MECO 35, Pont-a-Mousson 16.3.2010 Outline of the talk
More informationLecture 7. More dimensions
Lecture 7 More dimensions 67 68 LECTURE 7. MORE DIMENSIONS 7.1 Introduction In this lecture we generalize the concepts introduced so far to systems that evolve in more than one spatial dimension. While
More information02. Electric Field II
Universit of Rhode Island DigitalCommons@URI PHY 24: Elementar Phsics II Phsics Course Materials 215 2. Electric Field II Gerhard Müller Universit of Rhode Island, gmuller@uri.edu Creative Commons License
More informationCollective Effects. Equilibrium and Nonequilibrium Physics
Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, 2006 1 Collective Effects in Equilibrium and Nonequilibrium Physics Website: http://cncs.bnu.edu.cn/mccross/course/ Caltech
More informationFrom Particles to Fields
From Particles to Fields Tien-Tsan Shieh Institute of Mathematics Academic Sinica July 25, 2011 Tien-Tsan Shieh (Institute of MathematicsAcademic Sinica) From Particles to Fields July 25, 2011 1 / 24 Hamiltonian
More informationPhysics 443, Solutions to PS 2
. Griffiths.. Physics 443, Solutions to PS The raising and lowering operators are a ± mω ( iˆp + mωˆx) where ˆp and ˆx are momentum and position operators. Then ˆx mω (a + + a ) mω ˆp i (a + a ) The expectation
More informationThis work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 4.0 License.
University of Rhode Island DigitalCommons@URI Nonequilibrium Statistical Physics Physics Course Materials 215 8. Brownian Motion Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative Commons
More informationLecture notes for QFT I (662)
Preprint typeset in JHEP style - PAPER VERSION Lecture notes for QFT I (66) Martin Kruczenski Department of Physics, Purdue University, 55 Northwestern Avenue, W. Lafayette, IN 47907-036. E-mail: markru@purdue.edu
More informationCollective Effects. Equilibrium and Nonequilibrium Physics
1 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, 2006 1 Collective Effects in Equilibrium and Nonequilibrium Physics Website: http://cncs.bnu.edu.cn/mccross/course/
More informationIntroduction to Path Integrals
Introduction to Path Integrals Consider ordinary quantum mechanics of a single particle in one space dimension. Let s work in the coordinate space and study the evolution kernel Ut B, x B ; T A, x A )
More information2 Canonical quantization
Phys540.nb 7 Canonical quantization.1. Lagrangian mechanics and canonical quantization Q: How do we quantize a general system?.1.1.lagrangian Lagrangian mechanics is a reformulation of classical mechanics.
More informationADVANCED TOPICS IN THEORETICAL PHYSICS II Tutorial problem set 2, (20 points in total) Problems are due at Monday,
ADVANCED TOPICS IN THEORETICAL PHYSICS II Tutorial problem set, 15.09.014. (0 points in total) Problems are due at Monday,.09.014. PROBLEM 4 Entropy of coupled oscillators. Consider two coupled simple
More informationNPTEL
NPTEL Syllabus Nonequilibrium Statistical Mechanics - Video course COURSE OUTLINE Thermal fluctuations, Langevin dynamics, Brownian motion and diffusion, Fokker-Planck equations, linear response theory,
More informationDistributions of statistical mechanics
CHAPTER II Distributions of statistical mechanics The purpose of Statistical Mechanics is to explain the thermodynamic properties of macroscopic systems starting from underlying microscopic models possibly
More informationLecture 10 Planck Distribution
Lecture 0 Planck Distribution We will now consider some nice applications using our canonical picture. Specifically, we will derive the so-called Planck Distribution and demonstrate that it describes two
More informationPhys 622 Problems Chapter 5
1 Phys 622 Problems Chapter 5 Problem 1 The correct basis set of perturbation theory Consider the relativistic correction to the electron-nucleus interaction H LS = α L S, also known as the spin-orbit
More informationMicroscopic Hamiltonian dynamics perturbed by a conservative noise
Microscopic Hamiltonian dynamics perturbed by a conservative noise CNRS, Ens Lyon April 2008 Introduction Introduction Fourier s law : Consider a macroscopic system in contact with two heat baths with
More information1 The Quantum Anharmonic Oscillator
1 The Quantum Anharmonic Oscillator Perturbation theory based on Feynman diagrams can be used to calculate observables in Quantum Electrodynamics, like the anomalous magnetic moment of the electron, and
More informationConcepts for Specific Heat
Concepts for Specific Heat Andreas Wacker 1 Mathematical Physics, Lund University August 17, 018 1 Introduction These notes shall briefly explain general results for the internal energy and the specific
More informationNonperturbational Continued-Fraction Spin-offs of Quantum Theory s Standard Perturbation Methods
Nonperturbational Continued-Fraction Spin-offs of Quantum Theory s Standard Perturbation Methods Steven Kenneth Kauffmann Abstract The inherently homogeneous stationary-state and time-dependent Schrödinger
More information16 A Statistical Mechanical Theory of Quantum Dynamics in Classical Environments
16 A Statistical Mechanical Theory of Quantum Dynamics in Classical Environments Raymond Kapral 1 and Giovanni Ciccotti 2 1 Chemical Physics Theory Group, Department of Chemistry, University of Toronto,
More information561 F 2005 Lecture 14 1
56 F 2005 Lecture 4 56 Fall 2005 Lecture 4 Green s Functions at Finite Temperature: Matsubara Formalism Following Mahan Ch. 3. Recall from T=0 formalism In terms of the exact hamiltonian H the Green s
More information01. Introduction and Electric Field I
University of Rhode Island DigitalCommons@URI PHY 204: lementary Physics II Physics Course Materials 2015 01. Introduction and lectric Field I Gerhard Müller University of Rhode Island, gmuller@uri.edu
More information5.74 Introductory Quantum Mechanics II
MIT OpenCourseWare http://ocw.mit.edu 5.74 Introductory Quantum Mechanics II Spring 009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Andrei Tokmakoff,
More informationPhD in Theoretical Particle Physics Academic Year 2017/2018
July 10, 017 SISSA Entrance Examination PhD in Theoretical Particle Physics Academic Year 017/018 S olve two among the four problems presented. Problem I Consider a quantum harmonic oscillator in one spatial
More information03. Electric Field III and Electric Flux
Universit of Rhode Island DigitalCommons@URI PHY 204: lementar Phsics II Phsics Course Materials 2015 03. lectric Field III and lectric Flu Gerhard Müller Universit of Rhode Island, gmuller@uri.edu Creative
More information(a) Write down the total Hamiltonian of this system, including the spin degree of freedom of the electron, but neglecting spin-orbit interactions.
1. Quantum Mechanics (Spring 2007) Consider a hydrogen atom in a weak uniform magnetic field B = Bê z. (a) Write down the total Hamiltonian of this system, including the spin degree of freedom of the electron,
More informationOpen Quantum Systems and Markov Processes II
Open Quantum Systems and Markov Processes II Theory of Quantum Optics (QIC 895) Sascha Agne sascha.agne@uwaterloo.ca July 20, 2015 Outline 1 1. Introduction to open quantum systems and master equations
More informationGreen s functions: calculation methods
M. A. Gusmão IF-UFRGS 1 FIP161 Text 13 Green s functions: calculation methods Equations of motion One of the usual methods for evaluating Green s functions starts by writing an equation of motion for the
More informationwhere β = 1, H = H µn (3.3) If the Hamiltonian has no explicit time dependence, the Greens functions are homogeneous in time:
3. Greens functions 3.1 Matsubara method In the solid state theory class, Greens functions were introduced as response functions; they can be used to determine the quasiparticle density of states. They
More informationCHAPTER V. Brownian motion. V.1 Langevin dynamics
CHAPTER V Brownian motion In this chapter, we study the very general paradigm provided by Brownian motion. Originally, this motion is that a heavy particle, called Brownian particle, immersed in a fluid
More informationTheory of metallic transport in strongly coupled matter. 2. Memory matrix formalism. Andrew Lucas
Theory of metallic transport in strongly coupled matter 2. Memory matrix formalism Andrew Lucas Stanford Physics Geometry and Holography for Quantum Criticality; Asia-Pacific Center for Theoretical Physics
More informationWeek 5-6: Lectures The Charged Scalar Field
Notes for Phys. 610, 2011. These summaries are meant to be informal, and are subject to revision, elaboration and correction. They will be based on material covered in class, but may differ from it by
More informationDamped Harmonic Oscillator
Damped Harmonic Oscillator Wednesday, 23 October 213 A simple harmonic oscillator subject to linear damping may oscillate with exponential decay, or it may decay biexponentially without oscillating, or
More informationThis work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 4.0 License.
University of Rhode Island DigitalCommons@URI PHY 204: Elementary Physics II Physics Course Materials 2015 16. Faraday's Law Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative Commons
More informationPOSTULATES OF QUANTUM MECHANICS
POSTULATES OF QUANTUM MECHANICS Quantum-mechanical states - In the coordinate representation, the state of a quantum-mechanical system is described by the wave function ψ(q, t) = ψ(q 1,..., q f, t) (in
More informationMajor Concepts Langevin Equation
Major Concepts Langevin Equation Model for a tagged subsystem in a solvent Harmonic bath with temperature, T Friction & Correlated forces (FDR) Markovian/Ohmic vs. Memory Chemical Kinetics Master equation
More informationChapter 1. Principles of Motion in Invariantive Mechanics
Chapter 1 Principles of Motion in Invariantive Mechanics 1.1. The Euler-Lagrange and Hamilton s equations obtained by means of exterior forms Let L = L(q 1,q 2,...,q n, q 1, q 2,..., q n,t) L(q, q,t) (1.1)
More information(Dynamical) quantum typicality: What is it and what are its physical and computational implications?
(Dynamical) : What is it and what are its physical and computational implications? Jochen Gemmer University of Osnabrück, Kassel, May 13th, 214 Outline Thermal relaxation in closed quantum systems? Typicality
More informationClassical and quantum simulation of dissipative quantum many-body systems
0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 Classical and quantum simulation of dissipative quantum many-body systems
More informationNon-Markovian Quantum Dynamics of Open Systems: Foundations and Perspectives
Non-Markovian Quantum Dynamics of Open Systems: Foundations and Perspectives Heinz-Peter Breuer Universität Freiburg QCCC Workshop, Aschau, October 2007 Contents Quantum Markov processes Non-Markovian
More information05. Electric Potential I
University of Rhode Island DigitalCommons@URI PHY 204: Elementary Physics II Physics Course Materials 2015 05. Electric Potential I Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative Commons
More informationNon-equilibrium time evolution of bosons from the functional renormalization group
March 14, 2013, Condensed Matter Journal Club University of Florida at Gainesville Non-equilibrium time evolution of bosons from the functional renormalization group Peter Kopietz, Universität Frankfurt
More informationG : Statistical Mechanics
G25.2651: Statistical Mechanics Notes for Lecture 15 Consider Hamilton s equations in the form I. CLASSICAL LINEAR RESPONSE THEORY q i = H p i ṗ i = H q i We noted early in the course that an ensemble
More informationLangevin Methods. Burkhard Dünweg Max Planck Institute for Polymer Research Ackermannweg 10 D Mainz Germany
Langevin Methods Burkhard Dünweg Max Planck Institute for Polymer Research Ackermannweg 1 D 55128 Mainz Germany Motivation Original idea: Fast and slow degrees of freedom Example: Brownian motion Replace
More informationQuantization of the Spins
Chapter 5 Quantization of the Spins As pointed out already in chapter 3, the external degrees of freedom, position and momentum, of an ensemble of identical atoms is described by the Scödinger field operator.
More information3. Greens functions. H = H 0 + V t. V t = ˆBF t. e βh. ρ 0 ρ t. 3.1 Introduction
3. Greens functions 3.1 Introduction Greens functions appear naturally as response functions, i.e. as answers to the function how a quantum mechanical system responds to an external perturbation, like
More informationAn Isometric Dynamics for a Causal Set Approach to Discrete Quantum Gravity
University of Denver Digital Commons @ DU Mathematics Preprint Series Department of Mathematics 214 An Isometric Dynamics for a Causal Set Approach to Discrete Quantum Gravity S. Gudder Follow this and
More informationCaltech Ph106 Fall 2001
Caltech h106 Fall 2001 ath for physicists: differential forms Disclaimer: this is a first draft, so a few signs might be off. 1 Basic properties Differential forms come up in various parts of theoretical
More informationThe Particle-Field Hamiltonian
The Particle-Field Hamiltonian For a fundamental understanding of the interaction of a particle with the electromagnetic field we need to know the total energy of the system consisting of particle and
More information