Institut für Theoretische Physik und Astrophysik Christian ...bonitz/si10/bonitz2.pdfDense plasma in planets, compact stars, quark-gluon plasma Electron gas in metals, electron-hole

IntroductionIntroduction to to thethe theorytheoryof of densedense plasmasplasmas

Michael BonitzInstitut für Theoretische Physik und Astrophysik

Christian-Albrechts-Universität zu Kiel

2nd Summer Institute „Complex Plasmas“, Greifswald, 4 August 2010

G. Dominique , A. Filinov, P. LaPlante, H. Baumgartner, H. Kählert, L. Rosenthal, P. Ludwig, J. Böning

C. Henning, K. Balzer, S. Bauch, T. Ott, M. Heimsoth, D. Hochstuhl, MB

Cooperations:

V. Filinov (Moscow),A. Piel, H. Kersten (Kiel), Yu. Lozovik, A. Melzer, H. Fehske (Greifswald), J.W. Dufty (Florida) , Z. Donko and P. Hartmann (Budapest)

Group

ContentsContents

1. Theoretical approaches to nonideal classical plasmas

1.1. Fluid models: - ground state- thermodynamic equilibrium- collective excitations

1.2. First principle simulations: - molecular dynamics- Monte Carlo

ContentsContents (2)(2)

2. Nonideal quantum plasmas

4. First principle quantum simulations

2.2. Quantum kinetic theory

2.1. The Schrödinger equation

2.3. Hartree Fock and Nonequilibrium Green functions

2.4. Quantum hydrodynamics

3. Atoms and molecules. Partially ionized plasmas

1. Theoretical approaches to nonideal classical plasmas

Electrolytic solutionsDusty plasmasIons in traps

∑ ∑∑= ≠=

−++=N

iji

N

jii

N

i

i rrUrVm

pH11

2

)(21)(

2

V: external potential, U: pair interaction

N identical particles: Hamilton function

WhenWhen isis a a plasmaplasma nonidealnonideal ??Ideal gas behavior of electrons and ions: when Coulomb interaction energy U

is much smaller than kinetic energy K

TNkK B23

=In thermodynamic equilibrium:

∑≠ −

=ji ji

ji

rree

U||2

1

Estimate mean interaction energy: 3/1−∝≡− nrrr ji3/1

2

nreU ∝∝

Degree of nonideality:T

nkTre

KU 3/12||

∝∝=Γ

Nonideal behavior at low temperature or high density

NonidealNonideal plasmasplasmas

Trapped ions

Tne

rTke

EU

BKIN

3/122||∝==Γ

Occurences:

1. Low temperature

3. High charge

Dusty plasmas

Semiconductors

2. High density

1≥Γ

1. Theoretical approachesto nonideal classical plasmas

1.1 1.1 FluidFluid modelsmodels

Examples: strongly correlated ultracold ions or dust particles in spherical trap (Yukawa balls) concentric shells

Ion crystal, M. Drewsen

„Yukawa ball“O. Arp et al.

( talks on Monday, Tuesday)

Goal: theory explaining shell formation, density distribution in cluster

Classical density functional theory

U(r): Energy in external confinement,2nd term: mean field energy3rd term: correlation energy

Problem: find density profile n(r) that minimizes E

Mean field theory (T=0)

Note: parabolic decay with r, sharp step at finite radius RCoulomb limit: n(r)=const

Radial density distribution

Deviations for large screening due to neglect of correlation energy

C. Henning et al., Phys. Rev. E 74, 056403 (2006)

Radial density profile at T>0

Solve hierarchy for equilibrium distribution functions (BBGKY)Simplest approximation: mean field theory, example: Coulomb interaction

Wrighton, Dufty, Kählert, Bonitz, PRE 80, 066405 (2009)

)}({)()()( /)(

rnVrUrUern

indeff

kTrU eff

+=

∝ −

Result: Boltzmann factor:

Include correlation effects at T>0

)()(

),(

21

221

rnrnF

rrc

ex

δδβδ

−

=

Kraeft/Bonitz, J. Phys. Conf. Ser. 35, 78 (2006)Wrighton, Dufty, Kählert, Bonitz, PRE 80, 066405 (2009) and Contrib. Plasma Phys. 50, 26 (2010)

g: pair distribution function(joint probability)

F_ex: excess freeenergy

lengthwaveDeBroglie:Λ

Adjusted hypernetted chain approximation

HNC

Excellent reproduction of exact Monte Carlo data (crosses)Wrighton, Dufty, Kählert, Bonitz, PRE 80, 066405 (2009) and Contrib. Plasma Phys. 50, 26 (2010)

„Adjusted HNC“,

FluidFluid theorytheory: : collectivecollective excitationsexcitations

• Response of nonideal plasma in trap to weak excitationdetermined by (dN) collective modes

Examples: collective rotation, oscillation (sloshing), „breathing“ mode

• accessible by molecular dynamics simulation and experimentally (talk by A. Melzer)

• Alternative approach: time-dependent fluid theoryfor Coulomb systems: Dubinfor Yukawa systems: Kählert (2010)

FluidFluid theorytheory: : collectivecollective excitationsexcitations

V: external potential, friction included

Solve by linearization around Yukawa ground state (cold fluid limit)

Kählert, Bonitz, Phys. Rev. E (2010)

Complicated due to space-dependent density

FluidFluid theorytheory: : collectivecollective mode mode spectrumspectrum

Kählert, Bonitz, Phys. Rev. E (2010)Talk by Hanno Kählert, Monday

1.2 First 1.2 First principleprinciple simulationssimulations

Classical systems: - equations of motion can be solved exactly forlarge particle numbers (millions)

- no restrictions with respect to interaction

I. Monte Carlo: - virtually exact thermodynamic properties- also: efficient tools for time-dependent processes(„kinetic MC“, see talk by Lasse Rosenthal, Thursday)

II. Molecular dynamics: - virtually exact equilibrium and nonequilibriumproperties (see talks by Peter Hartmann and Torben Ott, Tuesday)

simulations very useful in combination with analytical tools

Classical Monte Carlo • calculation of thermodynamic averages

• Probability distribution for canonical ensemble with

• high-dimensional integral, evaluation with Monte Carlo algorithm:

• create configurations with probability distribution p(R)• Markov-chain• Metropolis algorithm

Metropolis algorithm• transition probability between two configurations

• always accept steps that decrease potential energy• if ΔU>0 accept new state with probability given by Boltzmann factor

• satisfies detailed-balance condition

• calculate averages according to

For details see: A. Filinov and M. Bonitz „Classical and Quantum Monte Carlo“, Chapter in: „Introduction to Computational Methods in Many-Body Physics“, Rinton Press, Princeton 2006

Classical Monte Carlo: Examplesdensity profile of 3D Yukawa balls: n(r) pair-distribution function g(r)

Density profile of Coulomb balls:J. Wrighton, J.W. Dufty, H. Kählert, and M. Bonitz,Phys. Rev. E 80, 066405 (2009)And Contrib. Plasma Phys. 50, 26-30 (2010)

Probability of metastable states

temperature kT

H. Kählert, P. Ludwig, H. Baumgartner, M. Bonitz, D. Block, S. Käding, A. Melzer, and A. Piel, Phys. Rev. E 78, 036408 (2008)

Interaction makes life interesting

Dipl.-Phys. Patrick LudwigPromotionsvortragRostock, 12. Dezember 2008

Coulomb-Korrelationen in HalbleiternInteraction makes life interesting

Molecular Dynamics

numerical technique for simultaneously solving the (Newtonian) equations of motion of many-particle systems.

Advantages

complete phase-space informationexact solution of model systemcomparison with experimental data (dusty plasmas, colloids, ions in traps..) large systems solvable (N~104-106) ∆t

t1

tN

Torben Ott

Disadvantages

purely classical descriptiondepends on choice of model (forces)possibly computationally demanding




Molecular Dynamics

interaction with other particles

external potentiale.g., electric fields

6N equations need to be solved

T. Ott, P. Ludwig, H. Kählert, M. Bonitz: "Molecular dynamics simulations of dusty plasmas" in M. Bonitz, N. Horing, P. Ludwig (eds.) "Introduction to Complex Plasmas", Springer (2010)

straightforward conceptbut: sophistication required




Molecular Dynamics

1000 particles of an OCP: (a) microcrystalline, (b) strongly correlated (liquid-like), (c) weakly correlated (gas-like) . Shown are the trajectories over the course of 100 (a,b) and 20 (c) plasma periods.

a b c

Alder, Wainwright 1959: contact interaction

continuous potential

B.J. Alder and T.E. Wainwright, Journal of Chem. Phys. 31, 459 (1959)



Coulomb-Korrelationen in HalbleiternInteraction makes life interestingMolecular Dynamics

1. Finite Systems (Yukawa- oder Coulomb-Balls)

crystal structure1

short-time behaviour, time-dependent crystallization2

phase transition, melting behaviour3

normal modes4

development of and comparison with analytical models5

transition to macoscopic systems

1H. Baumgartner et al., CPP 47, 281-290 (2007) 5W. Kraeft, M. Bonitz: J. Phys: Conf. Series 35, 94 (2006) 2H. Kählert, M. Bonitz PRL 104, 015001 (2010) (2009) 6T. Ott, M. Bonitz: PRL 103, 1950013J. Böning et al. PRL 100, 113401 (2008) 7M. Bonitz et al., PRL, 105, 055002 (2010) 4C. Henning et al, PRE 76, 036404 (2007)

2. „Infinite“ Systems (extended dusty plasmas)

characterization of diffusion and other transport processes6

waves and fluctuation spectra7

influence of magnetic fields7

nature of phase transitioninput for analytical models such as QLCA

2. Theoretical approaches to nonideal quantum plasmas

Dense plasma in planets, compact stars, quark-gluon plasmaElectron gas in metals, electron-hole plasma in semiconductorsLaser plasmas, ion beam compressed plasmas etc.

∑ ∑∑= ≠=

−++∇

−=N

iji

N

jii

N

i

i rrUrVm

H11

22

)(21)(

2h

V: external potential, U: pair interaction

N identical particles: Hamilton operator

Also: all reaction processes require quantum treatment (of electrons)

WhenWhen isis a a plasmaplasma „„quantumquantum““ ??

Quantum wave length („extension“) of a particle (De Broglie):mvh

=λ

depends on particle mass and velocity

r λ

Tmkh Bπλ 2/=Example: thermodynamic equilibrium:

1. Particles are quantum if r≥λ

Quantum Quantum degeneracydegeneracy

transform ratio of length scales to densities: 31

rn ∝

2. Define quantum degeneracy parameter (dimensionless) :12

3

+=

snλχ

Classical plasma: 1<χ

Quantum plasma: 1>χ

3. Energy criterion: quantum plasma if: ( ) 3/222

32

nm

EkT F πh=<

Fermi energy

CorrelationCorrelation andand Quantum Quantum effectseffectsCoulomb Interaction: reerU baab /)( =

TkU B/⟩⟨≡Γ

StrongCoulomb

correlations

1=ΓDeBroglie

wave length

Tmkh Bπλ 2/=

r

Overlapof wave functions,

Spin effects

Quantum effects

λ

r=λ

13 == λχ n

175=Γ crystal

2.1 2.1 TheThe SchrSchröödingerdinger equationequation

Time-dependent Schrödinger equation (TDSE)

Full N-particle TDSE

indicates, that the N-particle wave function is eithersymmetric (+: bosons) or antisymmetric (-: fermions)

includes all N particle coordinates

Example 2-particle TDSE

Two-particle wave function

(i) Bosons: Ψ(x1,x2) = Ψ(x2,x1)

(ii) Fermions: Ψ(x1,x2) = -Ψ(x2,x1)

Example: two interacting bosons in potential well

repulsive interactionw(x1-x2)

Numerical solution of the TDSE

Limitations:Maximum of N=4 particles for trapped particlesMaximum of N=2 particles for continuum dynamics (scattering, ionization, etc.)

Crank-Nicolson procedure on spatial grid

See S. Bauch et. al in “Introduction to Complex Plasma Physics“, Springer, Berlin 2010

Discretization of space (Δx) and time (Δt)Employing boundary conditionsPropagating initial state Ψ0

Approximation of time evolution operator

leads to tridiagonal system of equations

for new wave function (n+1) at time t+ΔtCoefficients ai, bi, ci and ri depend only on Ψn

Application: Electron-Ion collisions in strong fields

Method: wave packet scattering, full solution of time-dependent Schrödinger equation

Energy absorption from laser field during collisionQuantum mechanics: absorption of single photons (peaks in kinetic energy spectrum)Classical mechanics: transfer of quiver motion to translational motion, significant cut-off energies

S. Bauch and M. Bonitz, Contr. Plas. Phys. 49, 558 (2009)

Angular distributions of scattered electrons

Angle-resolved kinetic energy spectrum of scattered electrons

Intensity of laser field:1015 W/cm²

Photon energy:ω = 0.2 a.u.

Initial k of wave packet:k = 1.0

Classical cut-offs

Method: 2D wave packet scattering

S. Bauch and M. Bonitz, Contr. Plas. Phys. 49, 558 (2009)

2.2 Quantum 2.2 Quantum KineticKinetic TheoryTheory

Two main approaches:

1. Method of reduced density operators (directgeneralization of classical kinetic theory)

2. Second quantization (based on field operatorsfor fermions or bosons)

For details see book: M. Bonitz, „Quantum Kinetic Theory“

Density operator

Quantum description for mixed states

KΨΨ ...1Superposition of all N-particle states(Solutions of N-particle Schrödinger eq.)

Definition of density operator: 1,1

=ΨΨ= ∑=

ρρ TrWK

iiii

Weights (real probabilities): ∑ =i

iW 1

Replaces classicle N-particle probability distribution

Kinetic Theory for quantum plasmas (1)

Classical plasma Quantum plasma

N-particle probability density (x=r,p):

∫ =1...),,...( 11 NNNN fdxdxxxf

Liouville equation:

0},{ =−∂∂

NNN fHft

With Poisson brackets:

∑=

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

−∂∂

∂∂

=N

i N

N

N

N

N

N

N

NNN r

fpH

pf

rHfH

1},{

N-particle density operator

1ˆ,ˆ ...1 =NNN Tr ρρ

Von Neumann equation:

[ ] 0ˆ,ˆ1ˆ =−∂∂

NNN Hit

ρρh

With commutator:

abbaba ˆˆˆˆ]ˆ,ˆ[ −=



One-particle probability density (x=r,p):

),...(...)( 1211 NNN xxfdxdxxf ∫=

Kinetic equation:

},{ 12111 VfIpf

rU

rfv

tf

=∂∂

∂∂

−∂∂

+∂∂

Mean field approximation: 2112 fff ≈

Linearization: fff δ+= )0(11

Yields Vlasov dielectric function

One-particle density operator:

NNTrF ρ̂ˆ...21 =

Quantum Kinetic equation:

[ ] }ˆ,{ˆ,ˆ1ˆ12111 FVIFH

iF

t=−

∂∂

h

Mean field approximation:2112

ˆˆˆ FFF ≈

Linearization: FFF ˆˆˆ )0(11 δ+=

Yields „Random phase approximation“( see Norman Horing‘s talk)




Efficient numerical solution of kineticEquation using PIC-MCC

[particle in cell method (for mean field part) plus Monte Carlo methods for collisions.]

},{ 12111 VfIpf

rU

rfv

tf

=∂∂

∂∂

−∂∂

+∂∂

Quantum Kinetic equation:

[ ] }ˆ,{ˆ,ˆ1ˆ12111 FVIFH

iF

t=−

∂∂

h


• So far there are no efficient quantumgeneralizations of PIC, manychallenges remain

• alternative approach: classical PIC plus (small) quantum corrections

• we have developed direct solutions of QK equations usingTime-dependent Hartree Fock and Nonequilibrium Green‘s functions

2.3 Hartree-Fock and Nonequilibrium Greens Functions

Time-dependent Hartree-Fock

Use approximate N-particle wave function in TDSE

Fermions: Slater determinant

with one-electron (molecular) orbitals , , ,and

Hartree-Fock equations (TDHF) – Coupled system of effective one-particle TDSEs

Hartree term Fock (exchange) term

one-electron Hamiltonian

one-particle density matrix

Nonequilibrium Green's functions (NEGF) ITo go beyond mean field (TDHF) use concept of second quantization

Field operators

Properties

Definition of the one-particle NEGF

Keldysh/Kadanoff-Baym equations

one-particle orbital

+ adjointequation

self-energy functional

interaction potential

Nonequilibrium Green's functions II

Examples of accessible time-dependent observables

One-electron density

Current density

Particle number

Total energy

chemical potential

inverse temperature(~ ground state)


Example: Formation of Wigner moleculesConsider N=5 strongly correlated electrons in a 2D parabolic trap

Isotropic confinement of frequency Coupling parameter in trap

Length scaleMolecular orbitals

Example: Formation of Wigner molecules

Consider N=5 strongly correlated electrons in an 2D parabolic trap

Isotropic confinement of frequency Coupling parameter

Length scale

Inverse temperature(~ ground state)

Molecular orbitals


Example: Formation of Wigner molecules

Consider N=6 strongly correlated electrons in an 2D parabolic trap

Isotropic confinement of frequency Coupling parameter

Length scale

Finite temperature

Molecular orbitals


Occupation numbers

Example: Electron-hole bilayer

Phase-diagram of an N=10 bilayer

Mass symmetryLayer separation

New J. Phys. 10, 083031 (2008)

2.4 Quantum Hydrodynamics

• D. Bohm 1952: quantum mechanics in terms of real amplitude and phase

• Time-dependent Schrödinger equation yields two equations of motion:

Probability density (norm) P=A^2 is conserved

Quantum Hydrodynamics (2)

• classical Newtonian dynamics with additional „Quantum potential Q“

• quantum probability density follows from ensemble of trajectories

• But: limited to pure states

Quantum Hydrodynamics (3)

Extension to mixed states: Madelung (1926), Bohm (1954)Use Density matrix Wigner distribution f(R,p,t)

Example: 1 particle. P: probability amplitude, v: mean velocity

Need closure of hydrodynamic equations: approximations

Effective Schrödinger equation

1,1

=ΨΨ= ∑=

ρρ TrWK

iiii

0,iiW δ=

Recall density matrix:

1.) Bose condensate:

Nonlinear Schrödinger equation for ground state (Gross-Pitaevskii & beyond)

2.) Spatially homogeneous fermions at T=0(Manfredi)

AAeAx jxik

jjj ==Ψ ,)(

Incoherent fermion system mapped onto coherent wave functionPoorly justified and not verified

Neglect of exchange and correlations. Application to plasma questionable

See Chapter by Bonitz et al. In: „Introduction to Complex Plasmas“, Springer 2010

3. Atoms and molecules. Partially ionized plasmas

• e-i-Plasma at low temperature: Coulomb attraction leads to bound electrons

formation of atoms, molecules, clusters etc.

• Examples: - Planet interiors,- Low-temperature laboratory plasmas

• dense plasma:Bound state properties modified bysurrounding charged particles (screening)

• note: bound states always require a quantum treatment

PartiallyPartially ionizedionized HydrogenHydrogenHydrogen conductivity

(laboratory experiment) in cm/ohm410

210

010

210 −

210 − 010410 − 410

Nellis et al.

+ Fortov et al.

T=3,000-10,000K

Existence of an insulator-metal transition?Is this related to change of degree of ionization (chemical composition)?

3.1. Thermodynamic theory of partially ionized plasmas

High-density plasmas in the Interior of Jovian planets(Jupiter, Saturn)

Hydrogen/helium plasmawith total electron density of up to 32410 −= cmn

Thermodynamic and transport properties depend on how many electronsare free and how many are bound in atoms, molecules, i.e. on the degree of ionization, the plasma chemical composition.

There exist two approaches: 1. chemical models and 2. Physical models (mostly computer simulations)

Chemical models of partially ionizeddense plasmas

Example: partially ionized and dissociated hydrogen: e, p, H, H_2

2HHHHpe

↔+↔+

22 HH μμ =

Hpe μμμ =+

I. Starting point: Choice of relevant particle species

inta

idaa μμμ +=II. Chemical potentials:

⎥⎦

⎤⎢⎣

⎡

+=

12ln

3

a

aaa s

nkT λμ

Ideal part: a) classical particles:

11)2( )],([

04

122/332

+= −

∞+ ∫ aa

anTE

saa e

EdEmn μβπ h

ii) quantum particles (fermions):

Basis: Quantum statistical theories, integral equations etc.

III. Interaction contributions of chemical potentials- charged particle interactions: e-e, e-p, p-p - neutral particle interactions: H-H, H2-H2, H-H2 - charged-neutral particle interactions:

Nonideal Saha equation

Chemical composition:

Percentage of ionized, atomicand molcular hydrogen(proton number fraction)

Schlanges, Bonitz, Chjan, Contrib. Plasma Phys. 35, 109 (1995)

Mass action law (Saha equation): )(1 intintint2/1~),( HpeB eeenTK

nn EI

A

i μμμββ −+−−=

For more details see book: Kremp et al., „Quantum Statistics of charged particles“,Springer 2003

What happens to atoms at high compression?

Effect 1: Destabilization of atom in plasma environment:- Screening of e-i attraction, reduced binding energy(ionization potential) many-body effect

Effect 2: Overlap of two atoms (electron wave functions) at high n:Tunneling of electrons from one atom to anotherquantum destabilization, tunnel (pressure) ionization of atoms, „Mott effect“at densities corresponding to 32410.., −≅∝ cmneiar B

Both effects occur simultaneously, are equally important

Both effects occur even at zero temperature,Temperature increase helps to destroy atoms, molecules

Mott Effect. Pressure IonizationIdeal Plasma Non ideal Plasma

Lowering of the continuum edge due to screening and quantum effectsreduces number and ionization energy of bound statesvanishing distinction between free and bound states

Nonideal plasma: reduced current conduction: due to Coulomb interaction and bound state formation

Ideal plasma: conductivity increases with density of charged particles

PhasesPhases of of twotwo--componentcomponent (TCP) (TCP) CoulombCoulomb systemssystems

Re E

kTT23

=3/11 n

ra

r e

B

se

∝=

ion crystal in e Fermi gas

Bonitz, Filinov, Levashov, Fortov, Fehske, Phys. Rev. Lett. 95, 235006 (2005)

ion quantumliquid

Problems of Chemical Models

- inconsistent treatment of chargesand neutrals (critical at Mott point)

- subdivision in free and boundparticles artificial!

- Exclusion of particles other thanchosen in the beginning

These problems are avoided in the „physical picture“

4. First principle quantumsimulations

Requirements for quantum simulations

A. Single-particle properties

- coordinate and momentum not measurable simultaneously(Heisenberg uncertainty)

- Quantum particle has finite extension- quantum particle may be in many states(superposition principle)

- free quantum particle diffuses with time nn Ex),(Ψ

Fermions Bosons

B. Many-particle properties

- Spin statistics, indistinguishability(symmetry/antisymmetry of N-particlewave function)Availability of a quantum state for oneparticle depends on the states of all otherparticles (even without interaction!)

Types of simulations

Equilibrium Nonequilibrium

- Monte Carlo- Equilibrium Molecular Dynamics

A. Extension of classical methods

B. Special quantum methods

- Exact diagonalization (CI)- Density Functional Theory (DFT)- Hartree-Fock- Multiconfiguration Hartree-Fock (MCHF)- Matsubara Green functions

- Schrödinger equation- Time-dependent DFT- Time-dependent Hartree-Fock- Time-dependent MCHF- Nonequilibrium Green functions

- Nonequilibrium Molecular Dynamics- Kinetic equations (see above)

4.1 Quantum Monte Carlo

- Generalization of classical Monte Carlo (several methods)

- In particular: using Richard Feynmans‘s „Path integral“ representation of quantum mechanics „PIMC“

- Very successful „first-principle“ approach, avoids model assumptions

For details, see text book „Introduction to Computational Methods forMany-body Systems“, Rinton Press Princeton 2006,

S. Bauch, K. Balzer, P. Ludwig, A. Filinov, and M. Bonitz: “Introduction to quantum plasma simulations“, in: "Introduction to Complex Plasmas", M. Bonitz, N. Horing, P. Ludwig (eds.), Springer (2010) ,

Idea of path integral Monte Carlo

MkTMkTVKH /1/,/1, ===+= βτβwith system hamiltonian:

K, V: kinetic and potential energy operators, M: (large) integer number

Key quantity: canonical density operator (unknown): He βρ −=

Feynman‘s idea: express in terms of high-temperature density operator which is known:

Idea of path integral Monte Carlo (2)

In coordinate representation:

R contains coordinates of all particles (3N dim vector)

each particle is represented byTrajectory from R to R‘ („path“)

Density operator contains weightedSum over all paths (superposition)

Scheme of path integral Monte Carlo

MC procedure:

Optimize particle position and pathShape of probability density

Example: 3 particles

Fluctuating probability density of 3 particles

Spin statistics in path integral Monte Carlo

For bosons (fermions) density matrix has to be (anti-)symmetric:

Perform (anti-)symmetrization:

This is realized by exchanging (connecting) paths of several particles:Example: 5 particles on a plain with exchange of 1 and 2

Applications of path integral Monte Carlo

2 bosons in a harmonic trap

Variation of confinement strengthand temperatureSimulation: Jens Böning

Mesoscopic electron crystal

Melting by compressionSimulation: Alexei FilinovPhys. Rev. Lett. 86, 3851 (2001)

Applications of path integral Monte Carlo

Partially ionized hydrogen plasma, T=10,000KSimulation: Vladimir Filinov

32010 −= cmn 321103 −⋅= cmn

T = 10,000 K, n = 3⋅1025 сm-3, ρ = 50.2 g/сm3

Filinov, Bonitz, Fortov, JETP Letters 72, 245 (2000)

- proton- electron

- electron

Proton crystallization in dense HydrogenProton crystallization in dense Hydrogen

1st-principlePath integralMonte Carlo simulation

SummarySummary (1) (1) Nonideal plasmas are omnipresent in nature (astrophysics,Dusty and low temperature plasma, condensed matter…

Nonideal quantum plasmas: difficult due to simultaneousquantum, spin and correlation effects

Quantum tools: Hartree-Fock, Green‘s functions, quantum kinetics,Equilibrium: First principle path-integral Monte Carlo

Classical plasmas: powerful first-principle methods (MD, MC)and accurate fluid approaches

SummarySummary (2)(2)For more details see our text books:

M. Bonitz, „Quantum Kinetic Theory“, Teubner 1998

„Introduction to Computational Methods for Many-Particle Systems“,M. Bonitz and D. Semkat (eds.), Rinton Press, Princeton 2006

„Introduction to Complex Plasmas“, Springer 2010

http://www.theo-physik.uni-kiel.de/~bonitz

SupportedSupported byby DFG via DFG via TransregioTransregio--SFBSFB Greifswald/Kiel Greifswald/Kiel „„Grundlagen Komplexer PlasmenGrundlagen Komplexer Plasmen““

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Institut für Theoretische Physik und Astrophysik Christian ...bonitz/si10/bonitz2.pdfDense plasma in planets, compact stars, quark-gluon plasma Electron gas in metals, electron-hole