Hamiltonian Dynamics Discretized methods for computing Hamiltonian evolution (a) Eulers Method, stepsize 0.3 (b)Modified Eulers Method, stepsize 0.3 (c)Leapfrog Method, stepsize 0.3 (d) Leapfrog Method, stepsize 1 position(q) position(q)
Hamiltonian Dynamics • Discretized methods for computing Hamiltonian evolution
MCMC from hamiltonian dynamics Canonical distribution Hamiltonian dynamics keeps canonical distribution Invariant P(a, p)=exp(heap/t)= exp(U(q)/T)exp(K(p)/T) In physics the canonical distribution is the distribution of particles over phase space(q, p) in an isolated system(microcanonical ensemble) (system with fixed particle numbers N, volume (fixed micro states)and energy Eo)
MCMC from Hamiltonian dynamics • Canonical distribution – Hamiltonian dynamics keeps canonical distribution invariant. – 𝑃 𝑞, 𝑝 = 1 𝑍 exp(−𝐻(𝑞, 𝑝)/𝑇) = 1 𝑍 exp(−𝑈 𝑞 /𝑇) exp(−𝐾 𝑝 /𝑇) – In physics, the canonical distribution is the distribution of particles over phase space 𝑞, 𝑝 in an isolated system (microcanonical ensemble) (system with fixed particle numbers 𝑁, volume (fixed micro states) and energy 𝐸0 )
MCMC from hamiltonian dynamics Canonical distribution Can be derived from equal probability principle The system has discrete states: Us,n=N! alsfsD max InΩ,s.t ∑ fs=N,〉fE=E Use stirlings formula and introduce lagrange multipliers max NIn n ∑1(E=8)-2E) nf+1+a+BEs)6=05∝exp(-阝E The system has continuous states: P(q,p)=P(z maxs=p(z)Inp(z) dz, s.t. p(z)dz=1,p(z)E(z)=E we can get p(z)a exp( BE(z))
MCMC from Hamiltonian dynamics • Canonical distribution – Can be derived from equal probability principle • The system has discrete states: 𝑓𝑠 , Ω = 𝑁!/(ς𝑠 𝑓𝑠 !) max 𝑓𝑠 ln Ω , s. t. 𝑠 𝑓𝑠 = 𝑁, 𝑠 𝑓𝑠𝐸𝑠 = 𝐸0 Use Stirling’s formula and introduce Lagrange multipliers: max {𝑓𝑠} 𝑁 ln 𝑁 − 𝑠 𝑓𝑠 ln 𝑓𝑠 − 𝛼 𝑠 𝑓𝑠 − 𝑁 − 𝛽 𝑠 𝑓𝑠𝐸𝑠 − 𝐸0 − ln 𝑓𝑠 + 1 + 𝛼 + 𝛽𝐸𝑠 𝛿𝑓𝑠 = 0, 𝑓𝑠 ∝ exp(−𝛽𝐸𝑠 ) • The system has continuous states: 𝑃 𝑞, 𝑝 = 𝑃(𝑧) max 𝑆 = �� �� ln 𝑝 𝑧 𝑑𝑧, s.t. = �𝑑� �� �� 1, �� = �� �� �� �� 0, we can get 𝑝 𝑧 ∝ exp −𝛽𝐸 𝑧
M CMC from Hamiltonian dynamics Canonical distribution Hamiltonian dynamics keeps canonical distribution invariant Proof: apply liouville's theorem Note:p(z)∝eH(z) ,q=0n,H,p=0H. oP==Z.p(z)=-2iqiaa p(z)-Xipiapip(2) ∑(OnH)p(z)(-a1H)-∑(-anH)p(2)(-an1H)
MCMC from Hamiltonian dynamics • Canonical distribution – Hamiltonian dynamics keeps canonical distribution invariant. Proof: apply Liouville’s theorem Note: 𝑝 𝑧 ∝ 𝑒 −𝐻 𝑧 , 𝑞ሶ 𝑖 = 𝜕𝑝𝑖 𝐻, 𝑝ሶ 𝑖 = 𝜕𝑞𝑖 𝐻. 𝜕𝜌 𝜕𝑡 = −𝑧ሶ⋅ 𝛻𝑝 𝑧 = − σ𝑖 𝑞ሶ 𝑖𝜕𝑞𝑖 𝑝 𝑧 − σ𝑖 𝑝ሶ 𝑖𝜕𝑝𝑖 𝑝 𝑧 = − σ𝑖 𝜕𝑝𝑖 𝐻 𝑝 𝑧 −𝜕𝑞𝑖 𝐻 − σ𝑖 −𝜕𝑞𝑖 𝐻 𝑝 𝑧 −𝜕𝑝𝑖 𝐻 = 0