1. Statistical Data Analysis: II
PART II
1. From trajectories to Probability Distribution Functions (PDF)
2. Sampling PDF with Particle Dynamics

2. Sampling

3. Sampling PDF’s with Particle Dynamics
The ergodic theorem highlights an equivalence/duality
between the Trajectory (Dynamics) versus PDF
descriptions (Statistics).
The question is how to sample a PDF using trajectories,
i.e. Particle Dynamics.
Since this is a major theme of Statistical Mechanics and
Stochastic Optimization (search in hyperdimensional
spaces), next we dig a bit deeper into this matter.

4. What is sampling?
A sequence of numbers whose frequency reflects the given PDF
with frequencies:
is drawn
times on a total of
Such that:

5. Monte Carlo Sampling 1. Draw at random in X space 2. Compute
3. Draw a random
4. If
: Accept
Else: Reject and try again
Repeat 1-4 until N acceptances
Q: How about efficiency? N/Ntry can be very poor <10%

6. Dynamic Sampling (Trajectories)
Generate a discrete trajectory
Such that the time spent “around”
Forms a sequence
Such that
where
Plus: All values are accepted!
Minus: The trajectory should visit all target space X

7. The Dirac Distribution
By convoluting with a given function, picks up a single value
Namely:

8. Dirac Transformation Rules
By convoluting with a given function, picks up a single value
Corresponding to zeros of the Dirac argument (crossing points):
T1:
More complicated:
T2:
Why? Because:

9. Dirac transformation rule
Given the transformation
The Dirac distribution transforms accordingly to:
Namely:
Useful identity:

10. Dynamical systems Consider a first-order equation of motion:
With initial conditions:
This is a mapping from x0 to x
parametrized by time
Solved by:
The PDF generated by the single trajectory
with Initial Condition x0 is:
Dirac score
Each crossing at x scores 1.
Based on Dirac Rule T2:

11. Dynamical systems
The corresponding PDF generated out of an ensemble of
Initial Conditions x0 is:
Dirac score:

12. Dynamic sampling via stochastic dynamics
By now, we have learned how to sample using
Deterministic trajectories .
Since initial conditions matter, we may need MANY
initial conditions, or a VERY LONG single trajectory
(if the system is ergodic)
Both can be inefficient.
How about using stochastic dynamics?

13. PDF evolution
The probability of finding a particle starting at x0 at t=t0
at position x at time t, is given by:
By averaging over initial conditions:
What is the evolution equation of this PDF?
Therefore P obeys a continuity equation:
Steady-state:
As expected!

14. Stochastic Particle Dynamics
We add noise: Stochastic Particle Dynamics
Noise obeys Fluctuation-Dissipation Theorem:
Each noise realization gives rise
to its own trajectory. No need to
change the initial condition.
The formal solution is (Stratonovich):

15. Averaging over noise
One can therefore define p(x;t) by integrating over both
Initial conditions AND/OR Noise Distribution (typically gaussian).
Noise can mitigate/eliminate averaging over initial conditions
Differential Ito calculus:
The mean is unaffected:
The variance picks up a diffusive component:

16. Fokker-Planck equation
Therefore P obeys now a continuity-diffusion equation,
(Fokker-Planck Equation):
Stationary (Equilibrium) distribution:
Where:
and
Is the Partition Function
Each of the PDF discussed so far can thus be linked to a
dynamic stochastic process each with its own potential
V(x) and diffusion D (temperature)!

17. Thermodynamic Partition Function
For a thermodynamic system, x is 6N-dimensional
vector, with N order Avogadro!
Computing the PF is generally impossible other than for
Gaussian ensembles (quadratic potential):
Main analytical technique: Saddle Point
Expand around local minima and perform the Gaussian integrals
Main numerical technique: Metropolis Monte Carlo
Perform a random walk in x-space. If the move lowers V(x)
It is accepted. If it raises V(x), it is accepted with probability
exp(-deltaV), deltaV>0. Also, Stochastic Particle Dynamics.

18. Molecular Dynamics Hamiltonian eqs of motion
Energy = Hamiltonian function
Closed orbit in phase-space (x,p)
Each orbit corresponds to a different value of E
Microcanonical:
Canonical:

19. Sampling PDF’s with Molecular Dynamics
If the system is ergodic, a single trajectory of span T
is equivalent to M replicas of span T/M
The deterministic particle
dynamics is not effective
in visiting large portions
of phase space.
Many initial conditions
may be needed…

20. Sampling PDF’s with Langevin Dynamics
If the system is ergodic, a single trajectory of span T
Is equivalent to M replicas of span T/M
The stochastic Langevin
Dynamics is more effective
In visiting larger portions
of phase space

21. Which one is best suited to what?
PDF versus SDE
We are presented with a dual, equivalent representation
of statistical phenomena:
Statistical PDF versus Stochastic Particle Dynamics (SPD).
Which one is best suited to what?
SPD is more direct, it provides the detailed dynamics leading
to the statistical distributions.
However it entails heavy statistical averaging.
Capturing rare events can be excruciating.
PDF’s give directly the noise-free statistical distribution.
However one must solve the FPE, which is generally less
straightforward, especially as the number of dimensions
increases. In d>3-6, SPD is the only option
(Molecular Dynamics and/or MonteCarlo methods)

22. Appendix: A note on ergodicity
Ergodic systems visit their entire phase-space X, so that the
time spent on a given region R within X is proportional to the
Size of that region (“measure” in math language)
Ergodic
Ergodicity is often assumed but hardly proven on rigorous grounds.
Notable exception: Sinai Billiard

23. Broken-Ergodicity Ergodicity can be broken by teh existence of
Additional invariants of motion, besides energy
Broken-ergodicity, the white region is dynamically unaccessible
Because it implies a change in the invariant
Ensemble average:
Many complex systems show broken ergodicity:
Glasses, proteins….
Usually associated with highly corrugated free-energy landscapes
Historically, broken ergodicity was first observed in the
famous Fermi-Pasta-Ulam-Tsingou problem (see pdf file)

24. Ergodic Theorem E R Time average of a given observable A(x): G O N
Ensemble average:
Brackets denote
average over Initial
Conditions.
First order dynamical system:

25. Assignements 1. Sample the Gaussian distribution with the first order
stochastic dynamics xdot=-k*x + noise (Langevin equation)
2. Optional for the enthusiast:
Compute the PDF generated by an harmonic oscillator by performing
ensemble averaging over a set of initial conditions (positions and
velocities) with the same energy E (micro-canonical ensemble).
Hint: you must bin (x,p) phase space…

26. End of Lecture