Bernard Krauth 2012
From Werner KRAUTH
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| __FORCETOC__ | __FORCETOC__ | ||
| - | '''E. P. Bernard, W. Krauth''' | + | E. P. Bernard, W. Krauth ''Addendum to ''Event-chain Monte Carlo algorithms for hard-sphere systems'' '' Phys. Rev. E 86, 017701 (2012) |
| - | ''''' Event-driven Monte Carlo algorithm for general potentials ''''' '''(Preprint ArXiv (2011)) ''' | + | |
| =Paper= | =Paper= | ||
| Line 7: | Line 6: | ||
| '''Abstract''': | '''Abstract''': | ||
| We extend the [[Bernard_Krauth_Wilson_2009|event-chain Monte Carlo algorithm]] from hard-sphere interactions to the micro- | We extend the [[Bernard_Krauth_Wilson_2009|event-chain Monte Carlo algorithm]] from hard-sphere interactions to the micro- | ||
| - | canonical ensemble (constant potential energy) for general potentials. This event-drivenMonte Carlo | + | canonical ensemble (constant potential energy) for general potentials. This event-driven Monte Carlo |
| algorithm is non-local, rejection-free, and allows for the breaking of detailed balance. The algorithm | algorithm is non-local, rejection-free, and allows for the breaking of detailed balance. The algorithm | ||
| uses a discretized potential, but its running speed is asymptotically independent of the discretization. | uses a discretized potential, but its running speed is asymptotically independent of the discretization. | ||
| Line 13: | Line 12: | ||
| directly in the continuum limit. | directly in the continuum limit. | ||
| - | [http://arxiv.org/pdf/1102.4094v1 Electronic version (from arXiv, original version)] | + | [http://arxiv.org/pdf/1111.6964 Electronic version (from arXiv)] |
| + | |||
| + | [http://pre.aps.org/abstract/PRE/v86/i1/e017701 Published version (subscription required)] | ||
| + | |||
| + | =Note= | ||
| + | In a subsequent [[Michel_Kapfer_Krauth_2013| 2014 paper in Journal of Chemical Physics, with Manon Michel and Sebastian Kapfer]] entitled '''Generalized event-chain Monte Carlo: Constructing rejection-free global-balance algorithms from infinitesimal steps''', we presented an improved event-chain algorithm for continuous potentials that really works in the continuum, and does not require the restriction to the micro-canonical ensemble. | ||
| =Illustration= | =Illustration= | ||
| - | [[Image:Event stepped move.jpg|200px|]] Here's an illustration of the algorithm for the [http://en.wikipedia.org/wiki/Tower_of_Hanoi Tower-of-Hanoi potential]. | + | [[Image:Event stepped move.jpg|left|100px|border|thumb|Move of one particle in the Event-driven MC algorithm]] Event-driven Monte Carlo displacement for four particles interacting via the [http://en.wikipedia.org/wiki/Tower_of_Hanoi Tower-of-Hanoi potential]. Particle ''i'' moves from its initial configuration until its collision |
| + | event with particle ''i´'' (potential of Eq. (1) with ''E'' = 1/4 and | ||
| + | ''E'' = 3/4). Only displacements at energy ''E'' are discounted | ||
| + | from ℓ. | ||
| <br clear="all" /> | <br clear="all" /> | ||
| - | =Algorithm= | + | =Implementations= |
| + | |||
| + | ==Naive Python implementation == | ||
| + | Here is a naive Python implementation of the event-driven Monte Carlo algorithm for the stepped Hanoi-tower potential. In this implementation, we first choose a particle (called "next_particle"), | ||
| + | and translate the entire configuration such that next_particle comes to lie at (0,0). We then try to | ||
| + | move it in the positive x direction, computing all the possible events on the line. Each event is | ||
| + | characterized by the x position, the signature (=/- 1) and the particle responsible for it. Events are listed in list "next_events", then sorted. Going through this list we find the event which would move | ||
| + | the energy beyond the current value. | ||
| - | Here is a naive Python implementation of the algorithm | ||
| #!/usr/bin/python | #!/usr/bin/python | ||
| ###========+=========+=========+=========+=========+=========+=========+= | ###========+=========+=========+=========+=========+=========+=========+= | ||
| Line 199: | Line 212: | ||
| pylab.show() | pylab.show() | ||
| + | ==Comparison with direct-sampling algorithm== | ||
| + | Here is, a completely trivial constant-energy direct-sampling algorithm for four particles in a periodic square box. In the below figure, we compare its integrated pair-correlation function with the one of the event-driven algorithm for k = 100 (delta_E = 0.01). Outputs are truly the same. | ||
| + | |||
| + | |||
| + | ###========+=========+=========+=========+=========+=========+=========+= | ||
| + | ## PROGRAM : direct_stepped.py | ||
| + | ## TYPE : Python script (python 2.7) | ||
| + | ## PURPOSE : Direct sampling simulation for N particles with a stepped | ||
| + | ## potential in a square with periodic boundary conditions. | ||
| + | ## COMMENT : L is a list of tuples | ||
| + | ##========+=========+=========+=========+=========+=========+=========+ | ||
| + | from random import uniform, randint | ||
| + | import math, pylab, cPickle | ||
| + | box=4. | ||
| + | N=4 | ||
| + | def dist(x,y): | ||
| + | """periodic distance between two two-dimensional points x and y""" | ||
| + | d_x= abs(x[0]-y[0])%box | ||
| + | d_x = min(d_x,box-d_x) | ||
| + | d_y= abs(x[1]-y[1])%box | ||
| + | d_y = min(d_y,box-d_y) | ||
| + | return math.sqrt(d_x**2 + d_y**2) | ||
| + | def potential(r,k): | ||
| + | """particles have a radius of interaction sigma = 1. The interaction passes | ||
| + | in k steps from 1 (at r=1) to zero (at r=0). Conditions on interaction range | ||
| + | and box size | ||
| + | 2 sigma < box | ||
| + | assure that two particles interact only through one path.""" | ||
| + | if r > sigma: | ||
| + | pot = 0 | ||
| + | else: | ||
| + | pot = k - int(r*k) | ||
| + | return pot | ||
| + | def Energy_calc(L): | ||
| + | """elementary calculation of the energy for the configuration in list L """ | ||
| + | Energy = 0 | ||
| + | for k in range(1,N): | ||
| + | for l in range(k): | ||
| + | Energy += potential(dist(L[k],L[l]),Nstep) | ||
| + | return Energy | ||
| + | sigma= 1.0 | ||
| + | Nstep = 100 | ||
| + | energy=0.904 | ||
| + | Energy_hist = int(energy*Nstep+0.5) | ||
| + | data=[] | ||
| + | for iter in xrange(1000000): | ||
| + | L = [(uniform(0.,box),uniform(0.,box)) for k in range(N)] | ||
| + | Energy=Energy_calc(L) | ||
| + | if Energy == Energy_hist: | ||
| + | i=randint(0,N-1) | ||
| + | j=(i+randint(1,N-1))%N | ||
| + | data.append(dist(L[i],L[j])) | ||
| + | f = open('Direct_stepped'+str(Nstep)+'.data',"w") | ||
| + | cPickle.dump(data,f) | ||
| + | f.close() | ||
| + | pylab.hist(data,bins=40,normed=True,facecolor='green') | ||
| + | pylab.title("Direct_stepped N= "+str(N)+", k = "+str(Nstep)+" Energy = "+str(Energy_hist/float(Nstep))) | ||
| + | pylab.savefig('Direct_stepped'+str(Nstep)+'.png') | ||
| + | pylab.show() | ||
| + | |||
| + | [[Image:Direct stepped event stepped k 100.png|left|100px|border|thumb|Comparison of the Event-driven MC algorithm with a trivial direct-sampling]] Here is a comparison, at energy 0.904 of the integrated pair distribution function between the Event-driven MC algorithm and the trivial direct-sampling method. There is no doubt that the distributions are the same. | ||
| + | <br clear="all" /> | ||
| + | ==Sophisticated Python implementation== | ||
| + | |||
| + | As discussed in our paper, the event-drivent algorithm can be implemented for arbitrarily | ||
| + | small values of ''Delta_E'' with a complexity ''O''(1). This means that | ||
| + | the running time of the algorithm is independent of the discretization. | ||
| + | |||
| + | The below program implements the corresponding algorithm in basic Python. Running times were | ||
| + | about 6 minutes, for a 100-step discretization just as for a million steps. A more detailed description | ||
| + | of this program has unfortunately not been published yet. | ||
| + | |||
| + | [[Image:Event stepped 100 1000000.png|left|100px|border|thumb|Comparison of the Event-driven MC algorithm with different discretizations]] Here is a comparison, at energy 0.900 of the integrated pair distribution function between the Event-driven MC algorithm with different values of the discretization. Running times were roughly the same although the Hanoi potential was put together from 10^6 little plates. | ||
| + | <br clear="all" /> | ||
| + | ###========+=========+=========+=========+=========+=========+=========+= | ||
| + | ## PROGRAM : event_stepped_cont.py | ||
| + | ## TYPE : Python script (python 2.7) | ||
| + | ## PURPOSE : Improved event-chain simulation for N particles with a stepped | ||
| + | ## potential in a square with periodic boundary conditions. The program | ||
| + | ## keeps track of the continuous energy, as well as of the discrete one. | ||
| + | ## Running speed is roughly independent of the parameter N_step, which | ||
| + | ## governs the discretization of the potential | ||
| + | ## COMMENT : L is a list of tuples | ||
| + | ## VERSION : 08 Jan 2012 | ||
| + | ##========+=========+=========+=========+=========+=========+=========+ | ||
| + | from random import uniform, randint, choice, seed | ||
| + | import math, pylab, sys, cPickle,scipy.optimize | ||
| + | def translate(L,a): | ||
| + | """Translate configuration with respect to the vector a""" | ||
| + | L_trans=[((b[0] - a[0])%box,((b[1] - a[1])%box)) for b in L] | ||
| + | return L_trans | ||
| + | def dist(x,y): | ||
| + | """periodic distance between two two-dimensional points x and y""" | ||
| + | d_x= abs(x[0]-y[0])%box | ||
| + | d_x = min(d_x,box-d_x) | ||
| + | d_y= abs(x[1]-y[1])%box | ||
| + | d_y = min(d_y,box-d_y) | ||
| + | return math.sqrt(d_x**2 + d_y**2) | ||
| + | def potential(r): | ||
| + | """particles have a radius of interaction sigma = 1. | ||
| + | The interaction passes in N_steps steps from 1 (at r=1) to zero (at r=0). | ||
| + | Conditions on interaction range and box size | ||
| + | 2 sigma < box | ||
| + | assure that two particles interact only through one path.""" | ||
| + | if r > One: | ||
| + | pot = 0 | ||
| + | elif continuous == False: | ||
| + | pot = N_step - int(r*N_step) | ||
| + | else: | ||
| + | pot = One - r | ||
| + | return pot | ||
| + | def Energy(x,L,E_zero): | ||
| + | """energy of configuration L + [(x,Zero)], subtract zero energy""" | ||
| + | E = - E_zero | ||
| + | L_dummy = L + [(x,Zero)] | ||
| + | for k in range(1,len(L_dummy)): | ||
| + | for l in range(k): | ||
| + | E += potential(dist(L_dummy[k],L_dummy[l])) | ||
| + | return E | ||
| + | def deriv_potential(a,x): | ||
| + | """derivative of the uncropped continuum potential, with respect | ||
| + | to x.""" | ||
| + | return (a[0]-x)/dist(a,(x,Zero)) | ||
| + | def Energy_deriv(x,L_part): | ||
| + | """Derivative of the uncropped energy for particles in list L_part | ||
| + | with respect to (x,Zero). L_part contains interacting particles only""" | ||
| + | Deriv = 0 | ||
| + | for a in L_part: | ||
| + | Deriv += deriv_potential(a,x) | ||
| + | return Deriv | ||
| + | def x_image_calc(x_vec): | ||
| + | """x_vec describes the position of a particle inside the box. This | ||
| + | function computes the closest "image" of x_vec to the point (0,0). The | ||
| + | conditions on the box size box, the interaction range sigma and the | ||
| + | length of the interval l_max | ||
| + | l_max < box/2. | ||
| + | l_max + 2 sigma < box | ||
| + | assure that the image position is closest to the entire interval [(0,0) | ||
| + | (l_max,0).""" | ||
| + | x = x_vec[0] | ||
| + | y = x_vec[1] | ||
| + | if x < 0 or x > box or y < 0 or y > box: print 'error',sys.exit() | ||
| + | if x < box/2: x_image = x | ||
| + | else : x_image = x - box | ||
| + | if y < box/2: y_image = y | ||
| + | else: y_image = y-box | ||
| + | return (x_image, y_image) | ||
| + | def x_intersection(x_vec,d): | ||
| + | """compute intersections between the circle of radius d, with center x_vec, and the x-axis. | ||
| + | We suppose that abs(x_vec[1]) < d, so that two such intersections exist.""" | ||
| + | del_x= math.sqrt(d**2 - x_vec[1]**2) | ||
| + | return (x_vec[0]-del_x, x_vec[0]+del_x) | ||
| + | def Flip_conf(L,rot): | ||
| + | L_flip=[] | ||
| + | for (a,b) in L: | ||
| + | if rot == 1: L_flip.append((box - b,a)) | ||
| + | else: L_flip.append((b,box - a)) | ||
| + | return L_flip,-rot | ||
| + | def L_part_comp(x,L): | ||
| + | """compute the partial list of particles \in L which contribute to the | ||
| + | potential at (x,Zero)""" | ||
| + | L_part=[] | ||
| + | for a in L: | ||
| + | if (dist(a,(x,Zero)) < One): L_part.append(x_image_calc(a)) | ||
| + | return L_part | ||
| + | def next_event_stepped(x,L): | ||
| + | """Compute the next event, in the +x direction, for the configuration | ||
| + | L+[(x,0)], using the stepped energy.""" | ||
| + | intersections=[(float("inf"),1,1)] | ||
| + | for a in L: | ||
| + | b = x_image_calc(a) | ||
| + | if abs(b[1]) < One: | ||
| + | if b[0] > x: | ||
| + | dist_dummy = min(int((dist(b,(x,0)) - Event_eps)*N_step)/float(N_step),One) | ||
| + | if dist_dummy < abs(b[1]): dist_dummy += 1./N_step | ||
| + | else: | ||
| + | dist_dummy = min(int((dist(b,(x,0.)) + Event_eps)*N_step+1)/float(N_step),One) | ||
| + | a_dummy = x_intersection(b,dist_dummy) | ||
| + | if a_dummy[0]%box > x: intersections.append((a_dummy[0]%box, 1,a)) | ||
| + | if a_dummy[1]%box > x: intersections.append((a_dummy[1]%box,-1,a)) | ||
| + | intersections.sort() | ||
| + | next_event = intersections.pop(0) | ||
| + | return next_event | ||
| + | def next_event_continuous(L,Max_distance,E_zero): | ||
| + | """Compute the next event, in the +x direction, for the configuration | ||
| + | L+[(x=0,0)], using the continuous energy""" | ||
| + | intersections=[Zero] | ||
| + | for a in L: | ||
| + | b = x_image_calc(a) | ||
| + | if abs(b[1]) < One: | ||
| + | a_dummy = x_intersection(b,One) | ||
| + | if a_dummy[0]%box < Max_distance: intersections.append(a_dummy[0]%box) | ||
| + | if a_dummy[1]%box < Max_distance: intersections.append(a_dummy[1]%box) | ||
| + | intersections.sort() | ||
| + | intersections.append(Max_distance) | ||
| + | intervals = [] | ||
| + | # | ||
| + | # here, we characterize the intervals with differentiable potent. | ||
| + | # | ||
| + | lower_bound = intersections.pop(0) | ||
| + | while intersections != []: | ||
| + | upper_bound = intersections.pop(0) | ||
| + | # | ||
| + | # here we define the interval and the contributing particles, | ||
| + | # | ||
| + | interval = [lower_bound,upper_bound] | ||
| + | midpoint = (lower_bound + upper_bound)/Two | ||
| + | lower_bound = upper_bound | ||
| + | L_part = L_part_comp(midpoint,L) | ||
| + | intervals.append((tuple(interval),L_part[:])) | ||
| + | x_zero = Max_distance | ||
| + | collision = False | ||
| + | for obb in intervals: | ||
| + | x_0 = obb[0][0]; x_1 = obb[0][1] | ||
| + | E_0 = Energy(x_0,L,E_zero); E_1 = Energy(x_1,L,E_zero) | ||
| + | E_deriv_0 = Energy_deriv(x_0,obb[1]) | ||
| + | E_deriv_1 = Energy_deriv(x_1,obb[1]) | ||
| + | if E_deriv_0 * E_deriv_1 > 0 and E_0 * E_1 < 0: | ||
| + | x_zero_test = scipy.optimize.bisect(Energy,x_0,x_1,args=(L,E_zero)) | ||
| + | if Energy_deriv(x_zero_test,obb[1]) > Zero: | ||
| + | collision = True | ||
| + | x_zero = x_zero_test | ||
| + | break | ||
| + | elif E_deriv_0 * E_deriv_1 < 0 and min(E_0, E_1) < 0: | ||
| + | x_max = scipy.optimize.bisect \ | ||
| + | (Energy_deriv,x_0,x_1,args=obb[1]) | ||
| + | E_max = Energy(x_max,L,E_zero) | ||
| + | if E_max > 0 and E_0 < 0: | ||
| + | x_zero_test = scipy.optimize.bisect \ | ||
| + | (Energy,x_0,x_max,args=(L,E_zero)) | ||
| + | if Energy_deriv(x_zero_test,obb[1]) > Zero: | ||
| + | collision = True | ||
| + | x_zero = x_zero_test | ||
| + | break | ||
| + | elif E_max > 0 and E_0 > 0: | ||
| + | x_zero_test = scipy.optimize.bisect \ | ||
| + | (Energy,x_max,x_1,args=(L,E_zero)) | ||
| + | if Energy_deriv(x_zero_test,obb[1]) > Zero: | ||
| + | collision = True | ||
| + | x_zero = x_zero_test | ||
| + | break | ||
| + | return x_zero, collision, obb[1] | ||
| + | #============================================================================================== | ||
| + | #=========================== main program starts here ========================================= | ||
| + | #============================================================================================== | ||
| + | # | ||
| + | seed('Felix') | ||
| + | Event_eps = 1.e-12 # disallow events within too close bounds. | ||
| + | Zero = 0.0 | ||
| + | One = 1.0 | ||
| + | Two = 2.0 | ||
| + | box = 4.0 | ||
| + | sigma = 1.0 | ||
| + | N = 4 | ||
| + | N_step = 1000000 | ||
| + | E_zero = 0.9 | ||
| + | data = [] | ||
| + | continuous = False | ||
| + | while True: | ||
| + | L = [(uniform(0.,box),uniform(0.,box)) for k in range(N)] | ||
| + | a = choice(L) | ||
| + | L.remove(a) | ||
| + | L = translate(L,a) | ||
| + | Energy_level_initial = Energy(Zero,L,0) | ||
| + | L += [(Zero,Zero)] | ||
| + | if Energy_level_initial == int(E_zero*N_step): break | ||
| + | Energy_level = Energy_level_initial | ||
| + | rot = 1 | ||
| + | ltilde = 3.0/float(N_step) | ||
| + | for iter in xrange(200000): | ||
| + | if iter%1000 ==0: print iter, 'iter' | ||
| + | i=randint(0,N-1) | ||
| + | j=(i+randint(1,N-1))%N | ||
| + | data.append(dist(L[i],L[j])) | ||
| + | if randint(0,1) < 1: L,rot = Flip_conf(L,rot) | ||
| + | Zero_distance_to_go = ltilde | ||
| + | next_particle = choice(L) | ||
| + | # | ||
| + | # this iteration will be used up when the Zero_distance_to_go falls to zero | ||
| + | # | ||
| + | while Zero_distance_to_go > Zero: | ||
| + | Total_distance_to_go = min(box/2,box/2-sigma)*.3971 | ||
| + | L.remove(next_particle) | ||
| + | L = translate(L,next_particle) | ||
| + | Current_position = Zero | ||
| + | while min(Total_distance_to_go,Zero_distance_to_go) > Zero: | ||
| + | if Energy_level < int(N_step*E_zero - 6): | ||
| + | L = translate(L,(Current_position,Zero)) | ||
| + | Current_position = Zero | ||
| + | continuous = True | ||
| + | x_zero, collision, L_part = next_event_continuous(L,Total_distance_to_go,E_zero-6./N_step) | ||
| + | Current_position = x_zero | ||
| + | Total_distance_to_go -= x_zero | ||
| + | continuous = False | ||
| + | Energy_level = Energy(Current_position,L,0) | ||
| + | # | ||
| + | # this single-particle loop leads to either: | ||
| + | # 1. A particle collision | ||
| + | # 2. Zero_distance_to_go = 0 | ||
| + | # 3. Total_distance_to_go = 0 | ||
| + | # | ||
| + | next_position,next_energy_level,next_particle = next_event_stepped(Current_position,L) | ||
| + | distance_to_next_event = next_position - Current_position | ||
| + | if Energy_level < Energy_level_initial: | ||
| + | if Total_distance_to_go < distance_to_next_event: | ||
| + | Current_position += Total_distance_to_go | ||
| + | next_particle = (Current_position,Zero) | ||
| + | L.append(next_particle) | ||
| + | break | ||
| + | else: # go to event, but energy will not be too high: remain in loop of one particle... | ||
| + | Current_position += distance_to_next_event | ||
| + | Energy_level += next_energy_level | ||
| + | Total_distance_to_go -= distance_to_next_event | ||
| + | else: # Energy_level == Energy_level_intial | ||
| + | min_dist= min(Total_distance_to_go, Zero_distance_to_go) | ||
| + | if min_dist < distance_to_next_event: | ||
| + | Current_position += min_dist | ||
| + | Total_distance_to_go -= min_dist | ||
| + | Zero_distance_to_go -= min_dist | ||
| + | next_particle = (Current_position,Zero) | ||
| + | L.append(next_particle) | ||
| + | break # Tdtg or Zdtg are zero, but next particle is old one | ||
| + | else: | ||
| + | # | ||
| + | # New event. | ||
| + | # | ||
| + | Current_position += distance_to_next_event | ||
| + | Total_distance_to_go -= distance_to_next_event | ||
| + | Zero_distance_to_go -= distance_to_next_event | ||
| + | if next_energy_level < 0: # new event, but energy not too high, remain in inner loop | ||
| + | Energy_level += next_energy_level | ||
| + | else: # exit one-particle loop: use the next particle as given in the list... | ||
| + | L.append((Current_position,Zero)) | ||
| + | break | ||
| + | f=open("event_stepped"+str(N_step)+".data","w") | ||
| + | cPickle.dump(data,f) | ||
| + | f.close() | ||
| + | pylab.hist(data,bins=40,normed=True) | ||
| + | pylab.title("Event_stepped ltilde= "+str(ltilde)+" N= "+str(N)+" Energy = "+str(Energy_level)) | ||
| + | pylab.savefig('Event_stepped.png') | ||
| + | pylab.show() | ||
| - | [[Category:Publication]] [[Category:2011]] [[Category:Algorithm]] [[Category:Hard spheres]] [[Category:Two dimensions]] | + | [[Category:Publication]] [[Category:2012]] [[Category:Algorithm]] |
Current revision
E. P. Bernard, W. Krauth Addendum to Event-chain Monte Carlo algorithms for hard-sphere systems Phys. Rev. E 86, 017701 (2012)
Contents |
[edit]
Paper
Abstract: We extend the event-chain Monte Carlo algorithm from hard-sphere interactions to the micro- canonical ensemble (constant potential energy) for general potentials. This event-driven Monte Carlo algorithm is non-local, rejection-free, and allows for the breaking of detailed balance. The algorithm uses a discretized potential, but its running speed is asymptotically independent of the discretization. We implement the algorithm for the cut-off linear potential, and discuss its possible implementation directly in the continuum limit.
Electronic version (from arXiv)
Published version (subscription required)
[edit]
Note
In a subsequent 2014 paper in Journal of Chemical Physics, with Manon Michel and Sebastian Kapfer entitled Generalized event-chain Monte Carlo: Constructing rejection-free global-balance algorithms from infinitesimal steps, we presented an improved event-chain algorithm for continuous potentials that really works in the continuum, and does not require the restriction to the micro-canonical ensemble.
[edit]
Illustration
event with particle i´ (potential of Eq. (1) with E = 1/4 and
E = 3/4). Only displacements at energy E are discounted
from ℓ.
[edit]
Implementations
[edit]
Naive Python implementation
Here is a naive Python implementation of the event-driven Monte Carlo algorithm for the stepped Hanoi-tower potential. In this implementation, we first choose a particle (called "next_particle"), and translate the entire configuration such that next_particle comes to lie at (0,0). We then try to move it in the positive x direction, computing all the possible events on the line. Each event is characterized by the x position, the signature (=/- 1) and the particle responsible for it. Events are listed in list "next_events", then sorted. Going through this list we find the event which would move the energy beyond the current value.
#!/usr/bin/python
###========+=========+=========+=========+=========+=========+=========+=
## PROGRAM : event_stepped.py
## TYPE : Python script (python 2.7)
## PURPOSE : Event-chain simulation for N particles with a stepped
## potential in a square with periodic boundary conditions.
## COMMENT : L is a list of tuples
## VERSION : 29 Nov 2011
##========+=========+=========+=========+=========+=========+=========+
from random import uniform, randint, choice, seed
import math, pylab, sys, cPickle
Energy_check_iteration = 0
def translate(x,del_x):
x_trans=((x[0]-del_x[0])%box,((x[1]-del_x[1])%box))
return x_trans
def dist(x,y):
"""periodic distance between two two-dimensional points x and y"""
d_x= abs(x[0]-y[0])%box
d_x = min(d_x,box-d_x)
d_y= abs(x[1]-y[1])%box
d_y = min(d_y,box-d_y)
return math.sqrt(d_x**2 + d_y**2)
def potential(r,k):
"""particles have a radius of interaction sigma = 1. The interaction passes
in k steps from 1 (at r=1) to zero (at r=0). Conditions on interaction range
and box size
2 sigma < box
assure that two particles interact only through one path."""
if r > sigma:
pot = 0
else:
pot = k - int(r*k)
return pot
def Energy_calc(L):
"""elementary calculation of the energy for the configuration in list L """
Energy=0
for k in range(1,N):
for l in range(k):
Energy += potential(dist(L[k],L[l]),Nstep)
return Energy
def x_image_calc(x_vec):
"""x_vec describes the position of a particle inside the box. This
function computes the closest "image" of x_vec to the point (0,0). The
conditions on the box size box, the interaction range sigma and the
length of the interval l_max
l_max < box/2.
l_max + 2 sigma < box
assure that the image position is closest to the entire interval [(0,0)
(l_max,0)."""
x = x_vec[0]
y = x_vec[1]
if x < 0 or x > box or y < 0 or y > box: print 'error',sys.exit()
if x < box/2: x_image = x
else : x_image = x - box
if y < box/2: y_image = y
else: y_image = y-box
return (x_image, y_image)
def x_intersection(x_vec,d):
"""compute intersections between the circle of radius d, with center x_vec, and the x-axis.
We suppose that abs(x_vec[1]) < d, so that two such intersections exist."""
del_x= math.sqrt(d**2 - x_vec[1]**2)
return (x_vec[0]-del_x, x_vec[0]+del_x)
def Flip_conf(L,rot):
L_flip=[]
for (a,b) in L:
if rot == 1: L_flip.append((box - b,a))
else: L_flip.append((b,box - a))
return L_flip,-rot
def Energy_check(L,message_string):
global Energy_check_iteration,Energy_level
Energy_check_iteration += 1
a= Energy_calc(L)
if a == Energy_level:
return ('OK',a,Energy_check_iteration)
else:
return ('Error',a,Energy_level,Energy_check_iteration)
#==============================================================================================
#=========================== main program starts here =========================================
#==============================================================================================
#
seed('Test')
Event_eps=1.e-12 # disallow events within too close bounds.
Zero = 0.0
One = 1.0
box = 4.0
sigma = 1.0
N = 4
Nstep = 100
data = []
while True:
L = [(uniform(0.,box),uniform(0.,box)) for k in range(N)]
Energy_level_initial = Energy_calc(L)
if Energy_level_initial == 90: break
Energy_level = Energy_level_initial
rot = 1
ltilde = 0.03
for iter in xrange(100000):
L_save = L[:]
if iter % 10000 ==0: print iter
i=randint(0,N-1)
j=(i+randint(1,N-1))%N
data.append(dist(L[i],L[j]))
if randint(0,1) < 1: L,rot = Flip_conf(L,rot)
#
# this iteration will be used up when the Zero_distance_to_go falls to zero
#
Zero_distance_to_go = ltilde
next_particle = choice(L)
while Zero_distance_to_go > Zero:
Total_distance_to_go = min(box/2,box/2-sigma)*.3971
L.remove(next_particle)
L = [translate(x,next_particle) for x in L]
next_events = []
Current_position = Zero
for x in L:
x_image = x_image_calc(x)
for k in range(1,Nstep+1):
distance = k/float(Nstep)
if abs(x_image[1]) < distance:
x_dummy = x_intersection(x_image,distance)
if x_dummy[0] > Event_eps and x_dummy[0] < Total_distance_to_go:
next_events.append((x_dummy[0],1,x))
if x_dummy[1] > Event_eps and x_dummy[1] < Total_distance_to_go:
next_events.append((x_dummy[1],-1,x))
next_events.append((float("inf"),0,(float("inf"),Zero))) # Final event, at infinity
next_events.sort(reverse=True)
while min(Total_distance_to_go,Zero_distance_to_go) > Zero:
#
# this single-particle loop leads to either:
# 1. A particle collision
# 2. Zero_distance_to_go = 0
# 3. Total_distance_to_go = 0
#
next_position,next_energy_level,next_particle = next_events.pop()
distance_to_next_event = next_position - Current_position
if Energy_level < Energy_level_initial:
if Total_distance_to_go < distance_to_next_event:
Current_position += Total_distance_to_go
next_particle = (Current_position,Zero)
L.append(next_particle)
break
else: # go to event, but energy will not be too high: remain in loop of one particle...
Current_position += distance_to_next_event
Energy_level += next_energy_level
Total_distance_to_go -= distance_to_next_event
else: # Energy_level == Energy_level_intial
min_dist= min(Total_distance_to_go, Zero_distance_to_go)
if min_dist < distance_to_next_event:
Current_position += min_dist
Total_distance_to_go -= min_dist
Zero_distance_to_go -= min_dist
next_particle = (Current_position,Zero)
L.append(next_particle)
break # we can break here, because either Tdtg or Zdtg are zero, but next particle is old one
else:
#
# We go to a new event.
#
Current_position += distance_to_next_event
Total_distance_to_go -= distance_to_next_event
Zero_distance_to_go -= distance_to_next_event
if next_energy_level < 0: # we went to event, but energy is not too high, remain in loop of one particle
Energy_level += next_energy_level
else: # exit one-particle loop: use the next particle as given in the list...
L.append((Current_position,Zero))
break
if Energy_calc(L) != 90:
L = L_save[:]
print 'error at ', iter
f=open("event_stepped_13.data","w")
cPickle.dump(data,f)
f.close()
pylab.hist(data,bins=40,normed=True)
pylab.title("Event_stepped ltilde= "+str(ltilde)+" N= "+str(N)+" Energy = "+str(Energy_level))
pylab.savefig('Event_stepped.png')
pylab.show()
[edit]
Comparison with direct-sampling algorithm
Here is, a completely trivial constant-energy direct-sampling algorithm for four particles in a periodic square box. In the below figure, we compare its integrated pair-correlation function with the one of the event-driven algorithm for k = 100 (delta_E = 0.01). Outputs are truly the same.
###========+=========+=========+=========+=========+=========+=========+=
## PROGRAM : direct_stepped.py
## TYPE : Python script (python 2.7)
## PURPOSE : Direct sampling simulation for N particles with a stepped
## potential in a square with periodic boundary conditions.
## COMMENT : L is a list of tuples
##========+=========+=========+=========+=========+=========+=========+
from random import uniform, randint
import math, pylab, cPickle
box=4.
N=4
def dist(x,y):
"""periodic distance between two two-dimensional points x and y"""
d_x= abs(x[0]-y[0])%box
d_x = min(d_x,box-d_x)
d_y= abs(x[1]-y[1])%box
d_y = min(d_y,box-d_y)
return math.sqrt(d_x**2 + d_y**2)
def potential(r,k):
"""particles have a radius of interaction sigma = 1. The interaction passes
in k steps from 1 (at r=1) to zero (at r=0). Conditions on interaction range
and box size
2 sigma < box
assure that two particles interact only through one path."""
if r > sigma:
pot = 0
else:
pot = k - int(r*k)
return pot
def Energy_calc(L):
"""elementary calculation of the energy for the configuration in list L """
Energy = 0
for k in range(1,N):
for l in range(k):
Energy += potential(dist(L[k],L[l]),Nstep)
return Energy
sigma= 1.0
Nstep = 100
energy=0.904
Energy_hist = int(energy*Nstep+0.5)
data=[]
for iter in xrange(1000000):
L = [(uniform(0.,box),uniform(0.,box)) for k in range(N)]
Energy=Energy_calc(L)
if Energy == Energy_hist:
i=randint(0,N-1)
j=(i+randint(1,N-1))%N
data.append(dist(L[i],L[j]))
f = open('Direct_stepped'+str(Nstep)+'.data',"w")
cPickle.dump(data,f)
f.close()
pylab.hist(data,bins=40,normed=True,facecolor='green')
pylab.title("Direct_stepped N= "+str(N)+", k = "+str(Nstep)+" Energy = "+str(Energy_hist/float(Nstep)))
pylab.savefig('Direct_stepped'+str(Nstep)+'.png')
pylab.show()
[edit]
Sophisticated Python implementation
As discussed in our paper, the event-drivent algorithm can be implemented for arbitrarily small values of Delta_E with a complexity O(1). This means that the running time of the algorithm is independent of the discretization.
The below program implements the corresponding algorithm in basic Python. Running times were about 6 minutes, for a 100-step discretization just as for a million steps. A more detailed description of this program has unfortunately not been published yet.
###========+=========+=========+=========+=========+=========+=========+=
## PROGRAM : event_stepped_cont.py
## TYPE : Python script (python 2.7)
## PURPOSE : Improved event-chain simulation for N particles with a stepped
## potential in a square with periodic boundary conditions. The program
## keeps track of the continuous energy, as well as of the discrete one.
## Running speed is roughly independent of the parameter N_step, which
## governs the discretization of the potential
## COMMENT : L is a list of tuples
## VERSION : 08 Jan 2012
##========+=========+=========+=========+=========+=========+=========+
from random import uniform, randint, choice, seed
import math, pylab, sys, cPickle,scipy.optimize
def translate(L,a):
"""Translate configuration with respect to the vector a"""
L_trans=[((b[0] - a[0])%box,((b[1] - a[1])%box)) for b in L]
return L_trans
def dist(x,y):
"""periodic distance between two two-dimensional points x and y"""
d_x= abs(x[0]-y[0])%box
d_x = min(d_x,box-d_x)
d_y= abs(x[1]-y[1])%box
d_y = min(d_y,box-d_y)
return math.sqrt(d_x**2 + d_y**2)
def potential(r):
"""particles have a radius of interaction sigma = 1.
The interaction passes in N_steps steps from 1 (at r=1) to zero (at r=0).
Conditions on interaction range and box size
2 sigma < box
assure that two particles interact only through one path."""
if r > One:
pot = 0
elif continuous == False:
pot = N_step - int(r*N_step)
else:
pot = One - r
return pot
def Energy(x,L,E_zero):
"""energy of configuration L + [(x,Zero)], subtract zero energy"""
E = - E_zero
L_dummy = L + [(x,Zero)]
for k in range(1,len(L_dummy)):
for l in range(k):
E += potential(dist(L_dummy[k],L_dummy[l]))
return E
def deriv_potential(a,x):
"""derivative of the uncropped continuum potential, with respect
to x."""
return (a[0]-x)/dist(a,(x,Zero))
def Energy_deriv(x,L_part):
"""Derivative of the uncropped energy for particles in list L_part
with respect to (x,Zero). L_part contains interacting particles only"""
Deriv = 0
for a in L_part:
Deriv += deriv_potential(a,x)
return Deriv
def x_image_calc(x_vec):
"""x_vec describes the position of a particle inside the box. This
function computes the closest "image" of x_vec to the point (0,0). The
conditions on the box size box, the interaction range sigma and the
length of the interval l_max
l_max < box/2.
l_max + 2 sigma < box
assure that the image position is closest to the entire interval [(0,0)
(l_max,0)."""
x = x_vec[0]
y = x_vec[1]
if x < 0 or x > box or y < 0 or y > box: print 'error',sys.exit()
if x < box/2: x_image = x
else : x_image = x - box
if y < box/2: y_image = y
else: y_image = y-box
return (x_image, y_image)
def x_intersection(x_vec,d):
"""compute intersections between the circle of radius d, with center x_vec, and the x-axis.
We suppose that abs(x_vec[1]) < d, so that two such intersections exist."""
del_x= math.sqrt(d**2 - x_vec[1]**2)
return (x_vec[0]-del_x, x_vec[0]+del_x)
def Flip_conf(L,rot):
L_flip=[]
for (a,b) in L:
if rot == 1: L_flip.append((box - b,a))
else: L_flip.append((b,box - a))
return L_flip,-rot
def L_part_comp(x,L):
"""compute the partial list of particles \in L which contribute to the
potential at (x,Zero)"""
L_part=[]
for a in L:
if (dist(a,(x,Zero)) < One): L_part.append(x_image_calc(a))
return L_part
def next_event_stepped(x,L):
"""Compute the next event, in the +x direction, for the configuration
L+[(x,0)], using the stepped energy."""
intersections=[(float("inf"),1,1)]
for a in L:
b = x_image_calc(a)
if abs(b[1]) < One:
if b[0] > x:
dist_dummy = min(int((dist(b,(x,0)) - Event_eps)*N_step)/float(N_step),One)
if dist_dummy < abs(b[1]): dist_dummy += 1./N_step
else:
dist_dummy = min(int((dist(b,(x,0.)) + Event_eps)*N_step+1)/float(N_step),One)
a_dummy = x_intersection(b,dist_dummy)
if a_dummy[0]%box > x: intersections.append((a_dummy[0]%box, 1,a))
if a_dummy[1]%box > x: intersections.append((a_dummy[1]%box,-1,a))
intersections.sort()
next_event = intersections.pop(0)
return next_event
def next_event_continuous(L,Max_distance,E_zero):
"""Compute the next event, in the +x direction, for the configuration
L+[(x=0,0)], using the continuous energy"""
intersections=[Zero]
for a in L:
b = x_image_calc(a)
if abs(b[1]) < One:
a_dummy = x_intersection(b,One)
if a_dummy[0]%box < Max_distance: intersections.append(a_dummy[0]%box)
if a_dummy[1]%box < Max_distance: intersections.append(a_dummy[1]%box)
intersections.sort()
intersections.append(Max_distance)
intervals = []
#
# here, we characterize the intervals with differentiable potent.
#
lower_bound = intersections.pop(0)
while intersections != []:
upper_bound = intersections.pop(0)
#
# here we define the interval and the contributing particles,
#
interval = [lower_bound,upper_bound]
midpoint = (lower_bound + upper_bound)/Two
lower_bound = upper_bound
L_part = L_part_comp(midpoint,L)
intervals.append((tuple(interval),L_part[:]))
x_zero = Max_distance
collision = False
for obb in intervals:
x_0 = obb[0][0]; x_1 = obb[0][1]
E_0 = Energy(x_0,L,E_zero); E_1 = Energy(x_1,L,E_zero)
E_deriv_0 = Energy_deriv(x_0,obb[1])
E_deriv_1 = Energy_deriv(x_1,obb[1])
if E_deriv_0 * E_deriv_1 > 0 and E_0 * E_1 < 0:
x_zero_test = scipy.optimize.bisect(Energy,x_0,x_1,args=(L,E_zero))
if Energy_deriv(x_zero_test,obb[1]) > Zero:
collision = True
x_zero = x_zero_test
break
elif E_deriv_0 * E_deriv_1 < 0 and min(E_0, E_1) < 0:
x_max = scipy.optimize.bisect \
(Energy_deriv,x_0,x_1,args=obb[1])
E_max = Energy(x_max,L,E_zero)
if E_max > 0 and E_0 < 0:
x_zero_test = scipy.optimize.bisect \
(Energy,x_0,x_max,args=(L,E_zero))
if Energy_deriv(x_zero_test,obb[1]) > Zero:
collision = True
x_zero = x_zero_test
break
elif E_max > 0 and E_0 > 0:
x_zero_test = scipy.optimize.bisect \
(Energy,x_max,x_1,args=(L,E_zero))
if Energy_deriv(x_zero_test,obb[1]) > Zero:
collision = True
x_zero = x_zero_test
break
return x_zero, collision, obb[1]
#==============================================================================================
#=========================== main program starts here =========================================
#==============================================================================================
#
seed('Felix')
Event_eps = 1.e-12 # disallow events within too close bounds.
Zero = 0.0
One = 1.0
Two = 2.0
box = 4.0
sigma = 1.0
N = 4
N_step = 1000000
E_zero = 0.9
data = []
continuous = False
while True:
L = [(uniform(0.,box),uniform(0.,box)) for k in range(N)]
a = choice(L)
L.remove(a)
L = translate(L,a)
Energy_level_initial = Energy(Zero,L,0)
L += [(Zero,Zero)]
if Energy_level_initial == int(E_zero*N_step): break
Energy_level = Energy_level_initial
rot = 1
ltilde = 3.0/float(N_step)
for iter in xrange(200000):
if iter%1000 ==0: print iter, 'iter'
i=randint(0,N-1)
j=(i+randint(1,N-1))%N
data.append(dist(L[i],L[j]))
if randint(0,1) < 1: L,rot = Flip_conf(L,rot)
Zero_distance_to_go = ltilde
next_particle = choice(L)
#
# this iteration will be used up when the Zero_distance_to_go falls to zero
#
while Zero_distance_to_go > Zero:
Total_distance_to_go = min(box/2,box/2-sigma)*.3971
L.remove(next_particle)
L = translate(L,next_particle)
Current_position = Zero
while min(Total_distance_to_go,Zero_distance_to_go) > Zero:
if Energy_level < int(N_step*E_zero - 6):
L = translate(L,(Current_position,Zero))
Current_position = Zero
continuous = True
x_zero, collision, L_part = next_event_continuous(L,Total_distance_to_go,E_zero-6./N_step)
Current_position = x_zero
Total_distance_to_go -= x_zero
continuous = False
Energy_level = Energy(Current_position,L,0)
#
# this single-particle loop leads to either:
# 1. A particle collision
# 2. Zero_distance_to_go = 0
# 3. Total_distance_to_go = 0
#
next_position,next_energy_level,next_particle = next_event_stepped(Current_position,L)
distance_to_next_event = next_position - Current_position
if Energy_level < Energy_level_initial:
if Total_distance_to_go < distance_to_next_event:
Current_position += Total_distance_to_go
next_particle = (Current_position,Zero)
L.append(next_particle)
break
else: # go to event, but energy will not be too high: remain in loop of one particle...
Current_position += distance_to_next_event
Energy_level += next_energy_level
Total_distance_to_go -= distance_to_next_event
else: # Energy_level == Energy_level_intial
min_dist= min(Total_distance_to_go, Zero_distance_to_go)
if min_dist < distance_to_next_event:
Current_position += min_dist
Total_distance_to_go -= min_dist
Zero_distance_to_go -= min_dist
next_particle = (Current_position,Zero)
L.append(next_particle)
break # Tdtg or Zdtg are zero, but next particle is old one
else:
#
# New event.
#
Current_position += distance_to_next_event
Total_distance_to_go -= distance_to_next_event
Zero_distance_to_go -= distance_to_next_event
if next_energy_level < 0: # new event, but energy not too high, remain in inner loop
Energy_level += next_energy_level
else: # exit one-particle loop: use the next particle as given in the list...
L.append((Current_position,Zero))
break
f=open("event_stepped"+str(N_step)+".data","w")
cPickle.dump(data,f)
f.close()
pylab.hist(data,bins=40,normed=True)
pylab.title("Event_stepped ltilde= "+str(ltilde)+" N= "+str(N)+" Energy = "+str(Energy_level))
pylab.savefig('Event_stepped.png')
pylab.show()
