Supplemental Materal: Causal Entropc Forces A. D. Wssner-Gross 1, 2, and C. E. Freer 3 1 Insttute for Appled Computatonal Scence, Harvard Unversty, Cambrdge, Massachusetts 02138, USA 2 The Meda Laboratory, Massachusetts Insttute of Technology, Cambrdge, Massachusetts 02139, USA 3 Department of Mathematcs, Unversty of Hawa at Manoa, Honolulu, Hawa 96822, USA 1
PARTICLE IN A BOX DETAILS The causal entropc force parameters used n the smulaton were τ = 10 s, T r = 4 10 5 K, and T c = 5 T r. The tme step used was ɛ = 0.025 s. The T c values n ths example and the others were chosen such that the observed behavors were exhbted over approxmately 500 tme steps. Snce the role of T c s to scale the magntude of the causal entropc force, the prmary result of smaller T c values s slower behavor. The degrees of freedom are summarzed n Table I. TABLE I. Degrees of freedom for partcle n a box. D.O.F. ( ) Forced? Mass (m ) q mn q max q (0) p (0) x Yes m 0 L L/10 0 y Yes m 0 L/5 L/10 0 The stochastc equatons of moton for the evoluton of path mcrostates (whch were sampled to calculate the causal entropc forces) were ṗ x (t) = g x (x(t), t) = p x ( t/ɛ ɛ)/ɛ + f x ( t/ɛ ɛ) + h x (x(t)) (1) ṗ y (t) = g y (x(t), t) = p y ( t/ɛ ɛ)/ɛ + f y ( t/ɛ ɛ) + h y (x(t)) (2) q x (t) = p x (t)/m (3) q y (t) = p y (t)/m, (4) and the determnstc equatons of moton for the evoluton of the causal macrostate (shown n Fgure 2(a) and Supplemental Move 1) when subected to the combned causal entropc force and expectaton energetc force contrbutons were ṗ x (t) = g x (x(t), t) + F x (t) = p x ( t/ɛ ɛ)/ɛ + F x (t) + h x (x(t)) (5) ṗ y (t) = g y (x(t), t) + F y (t) = p y ( t/ɛ ɛ)/ɛ + F y (t) + h y (x(t)) (6) q x (t) = p x (t)/m (7) q y (t) = p y (t)/m, (8) where g (x(t), t) represent the energetc force components defned n the man text, f (t) represent the random force components defned n the man text, F (t) represent the causal entropc force components defned n the man text, x, y represent the forced degrees of freedom of the partcle, 2
and h (x(t)) represent any nstantaneous perfectly elastc collson forces of the partcle wth hard walls of the unverse at q x = 0, L and q y = 0, L/5. The partcle mass was m = 10 21 kg. The allowed regon n system phase space was a rectangle wth wdth L = 400 m and heght L/5, as shown n Table I, wth momentum components bounded by p (t) m (q max q mn )/ɛ. CART AND POLE DETAILS The causal entropc force parameters used n the smulaton were τ = 25 s, T r = 4.0 10 6 K, and T c = 20 T r. The tme step used was ɛ = 0.05 s. The degrees of freedom are summarzed n Table II. TABLE II. Degrees of freedom for cart and pole. D.O.F. ( ) Forced? Mass (m ) q mn q max q (0) p (0) x Yes m 0 L L/10 0 θ No Ml 2 0 2π π 0 The stochastc equatons of moton for the evoluton of path mcrostates (whch were sampled to calculate the causal entropc forces) were ṗ x (t) m = g x(x(t), t) + ml q 2 θ (t) sn q θ(t) mg sn q θ (t) cos q θ (t) M + m + m cos 2 q θ (t) = p x( t/ɛ ɛ)/ɛ + f x ( t/ɛ ɛ) + h x (x(t)) + ml q 2 θ (t) sn q θ(t) mg sn q θ (t) cos q θ (t) M + m + m cos 2 q θ (t) ṗ θ (t) Ml = ṗx(t) cos q θ (t)/m + g sn q θ (t) 2 l q x (t) = p x (t)/m q θ (t) = p θ (t)/(ml 2 ), (9) (10) (11) (12) (13) and the determnstc equatons of moton for the evoluton of the causal macrostate (shown n Fgure 2(b) and Supplemental Move 2) when subected to the combned causal entropc force and 3
expectaton energetc force contrbutons were ṗ x (t) m = g x(x(t), t) + F x (t) + ml q 2 θ (t) sn q θ(t) mg sn q θ (t) cos q θ (t) M + m + m cos 2 q θ (t) = p x( t/ɛ ɛ) + F x (t) + h x (x(t)) + ml q 2 θ (t) sn q θ(t) mg sn q θ (t) cos q θ (t) M + m + m cos 2 q θ (t) ṗ θ (t) Ml = ṗx(t) cos q θ (t)/m + g sn q θ (t) 2 l q x (t) = p x (t)/m (14) (15) (16) (17) q θ (t) = p θ (t)/(ml 2 ), (18) where g (x(t), t) represent the energetc force components defned n the man text, f (t) represent the random force components defned n the man text, F (t) represent the causal entropc force components defned n the man text, x s the forced horzontal degree of freedom of the cart, θ s the unforced angular degree of freedom measured from the vertcal of the massless pole, h x (x(t)) represents any nstantaneous cart-wall collson forces, m = 10 21 kg s the mass at the end of the pole, M = 10 21 kg s the mass of the cart, g = 9.8 m/s 2 s the gravtatonal acceleraton, and l = 40 m s the pole length. The allowed regon n system phase space had wdth L = 400 m, as shown n Table II, wth momentum components bounded by p (t) m (q max q mn )/ɛ. Collsons of the cart wth the walls at q x = 0, L were perfectly nelastc, and the pole was allowed full angular freedom. TOOL USE PUZZLE DETAILS The causal entropc force parameters used n the smulaton were τ = 10 s, T r = 8.0 10 5 K, and T c = 1.25 T r. The tme step used was ɛ = 0.05 s. The degrees of freedom are summarzed n Table III. The tube had length 100 m and dameter 80 m. Dsc I (wth forced degrees of freedom x 1, y 1 ) had mass m 1 = m = 10 21 kg and radus 50 m, Dsc II (wth unforced degrees of freedom x 2, y 2 ) had mass m 2 = 0.5m and radus 20 m, and Dsc III (wth unforced degrees of freedom x 3, y 3 ) had mass m 3 = 2m and radus 20 m. The allowed regon n system phase space was a square of wdth L = 400 m, as shown n Table III, wth momentum components bounded by p (t) m (q max q mn )/ɛ. Dsc-dsc, dsc-wall, and dsc-tube collsons were all perfectly elastc. The stochastc equatons of moton for the evoluton of path mcrostates (whch were sampled 4
TABLE III. Degrees of freedom for tool use puzzle. D.O.F. ( ) Forced? Mass (m ) q mn q max q (0) p (0) x 1 Yes m 0 L 0.5L 0 y 1 Yes m 0 L 0.375L 0 x 2 No 0.5m 0 L 0.5L 0 y 2 No 0.5m 0 L 0.625L 0 x 3 No 2m 0 L 0.9L 0 y 3 No 2m 0 L 0.5L 0 to calculate the causal entropc forces) were ṗ x1 (t) = g x1 (x(t), t) = p x1 ( t/ɛ ɛ)/ɛ + f x1 ( t/ɛ ɛ) + h x1 (x(t)) (19) ṗ y1 (t) = g y1 (x(t), t) = p y1 ( t/ɛ ɛ)/ɛ + f y1 ( t/ɛ ɛ) + h y1 (x(t)) (20) ṗ x2 (t) = h x2 (x(t)) (21) ṗ y2 (t) = h y2 (x(t)) (22) ṗ x3 (t) = h x3 (x(t)) (23) ṗ y3 (t) = h y3 (x(t)) (24) q x1 (t) = p x1 (t)/m (25) q y1 (t) = p y1 (t)/m (26) q x2 (t) = p x2 (t)/(0.5m) (27) q y2 (t) = p y2 (t)/(0.5m) (28) q x3 (t) = p x3 (t)/(2m) (29) q y3 (t) = p y3 (t)/(2m), (30) and the determnstc equatons of moton for the evoluton of the causal macrostate (shown n Fgure 3 and Supplemental Move 3) when subected to the combned causal entropc force and 5
expectaton energetc force contrbutons were ṗ x1 (t) = g x1 (x(t), t) + F x1 (t) (31) = p x1 ( t/ɛ ɛ)/ɛ + F x1 (t) + h x1 (x(t)) (32) ṗ y1 (t) = g y1 (x(t), t) + F y1 (t) (33) = p y1 ( t/ɛ ɛ)/ɛ + F y1 (t) + h y1 (x(t)) (34) ṗ x2 (t) = h x2 (x(t)) (35) ṗ y2 (t) = h y2 (x(t)) (36) ṗ x3 (t) = h x3 (x(t)) (37) ṗ y3 (t) = h y3 (x(t)) (38) q x1 (t) = p x1 (t)/m (39) q y1 (t) = p y1 (t)/m (40) q x2 (t) = p x2 (t)/(0.5m) (41) q y2 (t) = p y2 (t)/(0.5m) (42) q x3 (t) = p x3 (t)/(2m) (43) q y3 (t) = p y3 (t)/(2m), (44) where g (x(t), t) represent the energetc force components defned n the man text, f (t) represent the random force components defned n the man text, F (t) represent the causal entropc force components defned n the man text, and h (x(t)) represent any nstantaneous dsc-dsc, dsc-wall, and dsc-tube collson forces. As control experments performed mplctly as part of the Monte Carlo path ntegral calculatons the model was stochastcally evolved 400 tmes from ts ntal confguraton for duraton τ. In only 15 out of 400 runs (3.75%) was such random evoluton able to release Dsc III from the tube such that q x3 (τ) q x3 (0) 0.125L or q y3 (τ) q y3 (0) 0.1L. SOCIAL COOPERATION PUZZLE DETAILS The causal entropc force parameters used n the smulaton were τ = 20 s, T r = 8.0 10 5 K, and T c = 2.5 T r. The tme step used was ɛ = 0.1 s. The degrees of freedom are summarzed n Table IV. 6
TABLE IV. Degrees of freedom for cooperaton puzzle. D.O.F. ( ) Forced? Mass (m ) q mn q max q (0) p (0) x 1 Yes m 0 0.5L 0.35L 0 y 1 Yes m 0.5L L 0.75L 0 x 2 Yes m 0.5L L 0.65L 0 y 2 Yes m 0.5L L 0.85L 0 x 3 No m 0.12L 0.88L 0.5L 0 y 3 No m 0 L 0.25L 0 y 4 No 0.1m 0 L 0.5L 0 Dsc I (wth forced degrees of freedom x 1, y 1 ) had radus 20 m and mass m 1 = m = 10 21 kg, Dsc II (wth forced degrees of freedom x 2, y 2 ) had radus 20 m and mass m 2 = m, Dsc III (wth unforced degrees of freedom x 3, y 3 ) had radus 80 m and mass m 3 = m, the handle dscs at the ends of the strng (wth poston parameterzed by unforced degree of freedom y 4 ) had radus 40 m and mass m 4 = 0.1m, and the strng had length 400 m. Dsc II was ntally postoned wth a vertcal offset of 0.1L from Dsc I. The allowed regon n system phase space s ndcated n Table IV, wth momentum components bounded by p (t) m (q max q mn )/ɛ. All dsc-dsc collsons were perfectly elastc (wth the excepton of perfectly nelastc dsc-handle collsons perpendcular to the strng), and all dsc-boundary collsons were perfectly nelastc. The vertcal degree of freedom of Dsc III (y 3 ) was subect to a drag force proportonal to velocty. Causal entropc forces on Dscs I and II were calculated ndependently from a common estmated global causal path dstrbuton, gvng nether dsc a detaled theory of mnd of the other. The stochastc equatons of moton for the evoluton of path mcrostates (whch were sampled 7
to calculate the causal entropc forces) were ṗ x1 (t) = g x1 (x(t), t) = p x1 ( t/ɛ ɛ)/ɛ + f x1 ( t/ɛ ɛ) + h x1 (x(t)) (45) ṗ y1 (t) = g y1 (x(t), t) = p y1 ( t/ɛ ɛ)/ɛ + f y1 ( t/ɛ ɛ) + h y1 (x(t)) (46) ṗ x2 (t) = g x2 (x(t), t) = p x2 ( t/ɛ ɛ)/ɛ + f x2 ( t/ɛ ɛ) + h x2 (x(t)) (47) ṗ y2 (t) = g y2 (x(t), t) = p y2 ( t/ɛ ɛ)/ɛ + f y2 ( t/ɛ ɛ) + h y2 (x(t)) (48) T(t) = 2h y 4 (x(t))m 3 /m 4 2h y3 (x(t)) 2m 3 /m 4 + 4 (49) ṗ x3 (t) = h x3 (x(t)) (50) ṗ y3 (t) = h y3 (x(t)) + 2T(t) D q y3 (t) (51) ṗ y4 (t) = h y4 (x(t)) T(t) (52) q x1 (t) = p x1 (t)/m (53) q y1 (t) = p y1 (t)/m (54) q x2 (t) = p x2 (t)/m (55) q y2 (t) = p y2 (t)/m (56) q x3 (t) = p x3 (t)/m (57) q y3 (t) = p y3 (t)/m (58) q y4 (t) = p y4 (t)/(0.1m), (59) and the determnstc equatons of moton for the evoluton of the causal macrostate (shown n Fgure 4 and Supplemental Move 4) when subected to the combned causal entropc force and 8
expectaton energetc force contrbutons were ṗ x1 (t) = g x1 (x(t), t) + F x1 (t) (60) = p x1 ( t/ɛ ɛ)/ɛ + F x1 (t) + h x1 (x(t)) (61) ṗ y1 (t) = g y1 (x(t), t) + F y1 (t) (62) = p y1 ( t/ɛ ɛ)/ɛ + F y1 (t) + h y1 (x(t)) (63) ṗ x2 (t) = g x2 (x(t), t) + F x2 (t) (64) = p x2 ( t/ɛ ɛ)/ɛ + F x2 (t) + h x2 (x(t)) (65) ṗ y2 (t) = g y2 (x(t), t) + F y2 (t) (66) = p y2 ( t/ɛ ɛ)/ɛ + F y2 (t) + h y2 (x(t)) (67) T(t) = 2h y 4 (x(t))m 3 /m 4 2h y3 (x(t)) 2m 3 /m 4 + 4 (68) ṗ x3 (t) = h x3 (x(t)) (69) ṗ y3 (t) = h y3 (x(t)) + 2T(t) D q y3 (t) (70) ṗ y4 (t) = h y4 (x(t)) T(t) (71) q x1 (t) = p x1 (t)/m (72) q y1 (t) = p y1 (t)/m (73) q x2 (t) = p x2 (t)/m (74) q y2 (t) = p y2 (t)/m (75) q x3 (t) = p x3 (t)/m (76) q y3 (t) = p y3 (t)/m (77) q y4 (t) = p y4 (t)/(0.1m), (78) where g (x(t), t) represent the energetc force components defned n the man text, f (t) represent the random force components defned n the man text, F (t) represent the causal entropc force components defned n the man text, h (x(t)) represent any nstantaneous dsc-dsc or dscboundary collson forces, D 0.1/ɛ s the drag coeffcent, and T s the strng tenson. As control experments performed mplctly as part of the Monte Carlo path ntegral calculatons the model was stochastcally evolved 1, 000 tmes from ts ntal confguraton for duraton τ. In only 39 out of 1, 000 runs (3.9%) was such random evoluton able to pull Dsc III to a poston accessble from ether of the compartments such that q y3 (τ) 0.3L. 9
PATH INTEGRAL CALCULATION DETAILS The causal entropc force path ntegral (11) can be calculated by a varety of technques. In the case of the example smulatons, we chose a Monte Carlo approach, as follows. Consder M stochastcally sampled paths x (t) startng from the ntal system state x(0), where the sample ndex s denoted by and the respectve ntal effectve fluctuatng force components are denoted by f (0), such that Pr(x (t) x(0)) > 0. For large M, we can estmate the condtonal path dstrbuton Pr(x (t) x(0)) as beng unform over a local patch of path-space around each path x (t) wth volume Ω [MPr(x (t) x(0))] 1, such that probablty s conserved: Ω Pr(x (t) x(0)) = 1. The path ntegral (11) can then be wrtten as F (X 0, τ) = 2T c T r 2T c T r = 2T c 1 T r M = 2T c T r 1 M = 2T c T r 1 M f (0) Pr(x(t) x(0)) ln Pr(x(t) x(0)) Dx(t) (79) x(t) f (0)(MΩ ) 1 ln(mω ) 1 Ω (80) f (0) ln(mω ) 1 = 2T c 1 T r M ln M f (0) + f (0) ln Ω (81) f (0) ln Ω = 2T c 1 T r M f (0) ln Ω ln Ω f (0) (82) Ω f (0) ln 2Tc 1 Ω Ω f (0) ln T r M Ω, (83) usng the fact that f (0) = 0. Note that the fnal form s expressed n terms of patch volume fractons Ω / Ω, whch can be calculated by kernel densty estmaton appled to the traectory set {x (t n )} over all samples. For reference, pseudocode for performng the partcle n a box smulaton usng ths Monte Carlo approach s shown below. Our full causal entropc force smulaton software wll be made avalable for exploraton at http://www.causalentropy.org. 10
Pseudocode for partcle n a box example functon CALCULATE_CAUSAL_ENTROPIC_FORCE(cur_macrostate) /* --- Monte Carlo path samplng --- */ sample_paths = EMPTY_PATH_LIST(); for = 1 to num_sample_paths do cur_path = EMPTY_PATH(); cur_state = cur_macrostate; for n = 0 to num_tme_steps do cur_path[n] = cur_state; cur_state = STEP_MICROSTATE(cur_state); sample_paths[] = cur_path; /* --- Kernel densty estmaton of log volume fractons --- */ log_volume_fracs = LOG_VOLUME_FRACTIONS(sample_paths); /* --- Sum force contrbutons --- */ force = ZERO_FORCE_VECTOR(); for = 1 to num_sample_paths do force += sample_paths[].ntal_force * log_volume_fracs[]; return 2 * (T_c / T_r) * (1.0 / num_sample_paths) * force; functon PERFORM_CAUSAL_ENTROPIC_FORCING(nt_macrostate) cur_macrostate = nt_macrostate; whle True do causal_entropc_force = CALCULATE_CAUSAL_ENTROPIC_FORCE(cur_macrostate); cur_macrostate = STEP_MACROSTATE(cur_macrostate,causal_entropc_force); PERFORM_CAUSAL_ENTROPIC_FORCING({"q_x":L/10, "q_y":l/10, "p_x":0, "p_y":0}); To whom correspondence should be addressed: alexwg@post.harvard.edu; http://www.alexwg.org Present address: Computer Scence and Artfcal Intellgence Laboratory, Massachusetts Insttute of Technology, Cambrdge, Massachusetts 02139, USA 11