Stationary Phase Monte Carlo Methods

Transkript

1 Stationary Phase Monte Carlo Methods Daniel Doro Ferrante G. S. Guralnik, J. D. Doll and D. Sabo HET Physics Dept, Brown University, USA.

2 Introduction: Motivations and Outline of the Method; phase transitions, smoothing of the action, Theory Behind; mollication/approimate identity method, Eamples and Results; Airy function action, φ 4 action, Conclusion.

3 Outline: The basic idea is to cast the partition function in such a way that its computation is manageable. Thus, the calculation of its derivatives (which yield the dierent Green's functions) become a feasible task, as well. In order to do so, the full [comple] eponential weight is convoluted with a probability-type function, so that its oscillations are smoothed out and a Monte Carlo calculation can be performed. This is particularly interesting when dealing with phase transitions since the factor ep(i S()) oscillates rapidly and these can be tamed via the smoothing procedure that will be described shortly so that a Monte Carlo computation can be performed. Also, coalescing stationary points are better handled by this method, again because of this smoothing technique. 3

4 Theory: The whole business is built upon the following construction: f() d f() ɛ d, () where with f() ɛ (f P ɛ )(), () = P ɛ () d =. P ɛ ( y) f(y) dy, In this case, the smoothing function is given by P ɛ () which is called a mollier or an approimate identity. The nice property that they have is: f() ɛ ɛ f() f() ɛ d ɛ f() d. 4

5 Simple eample: f m () = ep(i m ), P ɛ () = ep π ɛ ɛ f m () ɛ = ep i m i m ɛ i m ɛ Re[f] Im[mollf] m m 5

6 The method's application is in two steps:. Pre-averaging integrand: the generating functional is given of the by, Z ɛ [j] = e i S() i j ɛ [d] e i S() ɛ [d]. Thus, performing a saddle-point epansion: P ɛ (y) = ep yt (ɛ ) y (π) n det(ɛ ), I ɛ = f() e i S() i j ɛ [d] f( ) e i S( ) i j (3) ep BT ( + ɛ T S ɛ) B det( + ɛ T S ɛ) where ɛ is the covariance matri that denes the Gaussian distribution, B = ɛ S() and [S ] ij = S()/ i j (Hessian matri for S). 6 [d],

7 . Sampling (MC): better handle the Importance MC simulations. The equations become: W ɛ () = Z ɛ [j] = e i S() i j ɛ [dw ()] W () e i S() ɛ [dw ()] W (), (4) where W () is known as the importance function and the MC (Metropolis) sampling is done over this new measure. A reasonable choice for it is given by, W () = W ɛ () = e i S(). ɛ And, just as done before (saddle-point epansion), this becomes: ep i S( ) BT ( + ɛ T S ɛ) B det( + ɛ T S ɛ) 7

8 Eample: Airy Function The action is: S() = 3 /3 + t. Thus, Z[t] = ep i i t ep i 3 3 d d Ai(t) Ai() Plot of Re(e^(-S(,t))) t 5 5 Legend Re[ep(-I*s(, 6))] Re[ep(-I*s(y, 6))*P(-y,.3)] 8

9 Plot of W(, 6,e) e~ Legend W(, 6,.3) W(, 6,.8) W(, 6,.38) W(, 6,.68) 9

10 Eample: -dimensional φ 4 Theory The action is: S(φ) = µ φ / + g φ 4 /4. Thus, Z ɛ [j] = ep i µ φ + i g 4 φ4 i j φ ep i µ φ + i g4 φ4 ɛ ɛ dφ dφ Symmetric Phase (mu=m^>=; g gives higher order corrections) Broken Symmetric Phase -point Green s function at sqrt-mu/g mu.5 g.5 5 g mu

11 w(,,,e).5 e~ e+ 4e+ w(,,,e).5 e~ 4 <ep(-s(y,,,e))> y.5 e~ <ep(-s(y,,,e))> y.5 e~

12 Conclusion: This process can be interpreted as a way to implement block variables straight into the integrand, by means of smoothing it out. One big open question, under this view (coarse-graining, the graining parameter being ɛ), regards the renormalization procedure. to calculate! We have just begun