07-ilang-weight / 02-two-free-bodies
02two free bodiesverified
Determines the conserved quantities of two bodies and their field, whether a body acts on the other as strongly as it is acted upon, and how their accelerations are related to their inertias.
check_held_pair.py6 KB · raw
"""Check of item 2 (held bodies) of 07-ilang-weight/02-two-free-bodies.
N_f = 1 field object on a ring of L places, own term C_1 = -t_f sum (|x><x+1| + h.c.),
two held bodies acting as point terms g_1 P_{x1} + g_2 P_{x2}, g_i = sigma_i gamma_i M_i.
Compares the exact mutual term
M(x1,x2) = eps0(g1,g2) - eps0(g1,0) - eps0(0,g2) + eps0(0,0) (N_f = 1, n = 1/L)
with the closed forms (2.6)-(2.8) of the derivation, and checks translation, reflection and
exchange invariance (2.4), the non-product form at third order (2.9) and the highest-state map (2.5).
Output: output/held_pair.json and output/held_pair.log
"""
import json
from pathlib import Path
import numpy as np
OUT = Path(__file__).parent / "output"
OUT.mkdir(exist_ok=True)
def h_field(L, tf, pts):
H = np.zeros((L, L))
for x in range(L):
H[x, (x + 1) % L] -= tf
H[(x + 1) % L, x] -= tf
for x, g in pts:
H[x, x] += g
return H
def eps(L, tf, pts, top=False):
w = np.linalg.eigvalsh(h_field(L, tf, pts))
return w[-1] if top else w[0]
def mutual(L, tf, g1, g2, x1, x2, top=False):
return (eps(L, tf, [(x1, g1), (x2, g2)], top) - eps(L, tf, [(x1, g1)], top)
- eps(L, tf, [(x2, g2)], top) + eps(L, tf, [], top))
def lam_closed(L, d):
u = d * (L - d)
return (L**2 - 1) / (12 * L) - u / (2 * L)
def lam2_closed(L, d):
u = d * (L - d)
return (L**2 - 1) * (L**2 + 11) / (720 * L) - u / (12 * L) - u**2 / (24 * L)
def m2_closed(L, tf, g1, g2, d, n):
return -2 * n * g1 * g2 / tf * lam_closed(L, d)
def m3_closed(L, tf, g1, g2, d, n):
u = d * (L - d)
b3 = (20 * u**2 - 10 * (L**2 - 2) * u + (L**2 - 1) * (L**2 - 4)) / (60 * L**2)
return n / tf**2 * g1 * g2 * (g1 + g2) * b3
def run(L, tf, g1, g2):
n = 1.0 / L
res = {"L": L, "t_f": tf, "g1": g1, "g2": g2,
"kato_bound_|g1|+|g2|<": 2 * tf * np.sin(np.pi / L) ** 2}
# closed forms of Lambda^+ and (Lambda^+)^2 against the pseudo-inverse of the ring Laplacian
lap = h_field(L, 1.0, []) + 2 * np.eye(L)
lp = np.linalg.pinv(lap)
lp2 = lp @ lp
ds = np.arange(L)
res["max_err_Lambda_closed"] = float(np.max(np.abs(lp[0, ds] - lam_closed(L, ds))))
res["max_err_Lambda2_closed"] = float(np.max(np.abs(lp2[0, ds] - lam2_closed(L, ds))))
# exact mutual term for all position pairs
Mx = np.array([[mutual(L, tf, g1, g2, a, b) for b in range(L)] for a in range(L)])
Md = Mx[0]
res["max_dev_translation"] = float(np.max(np.abs(Mx - np.array(
[[Md[(b - a) % L] for b in range(L)] for a in range(L)]))))
res["max_dev_reflection"] = float(np.max(np.abs(Md - Md[(-ds) % L])))
Mswap = np.array([mutual(L, tf, g2, g1, 0, d) for d in ds])
res["max_dev_exchange"] = float(np.max(np.abs(Md - Mswap)))
# discrete forces F1 = -dM/dx1, F2 = -dM/dx2 (symmetric differences)
F1 = np.array([-(Mx[1, d] - Mx[L - 1, d]) / 2 for d in ds])
F2 = np.array([-(Mx[0, (d + 1) % L] - Mx[0, (d - 1) % L]) / 2 for d in ds])
res["max_|F1+F2|"] = float(np.max(np.abs(F1 + F2)))
# perturbative comparison at three scales of the couplings
scal = []
for s in [1.0, 0.5, 0.25]:
a1, a2 = s * g1, s * g2
Me = np.array([mutual(L, tf, a1, a2, 0, d) for d in ds])
Mm = np.array([mutual(L, tf, -a1, -a2, 0, d) for d in ds])
m2 = m2_closed(L, tf, a1, a2, ds, n)
m3 = m3_closed(L, tf, a1, a2, ds, n)
scal.append({"scale": s,
"max|M|": float(np.max(np.abs(Me))),
"max|M-M2|": float(np.max(np.abs(Me - m2))),
"max|M-M2-M3|": float(np.max(np.abs(Me - m2 - m3))),
"max|odd part - M3|": float(np.max(np.abs((Me - Mm) / 2 - m3)))})
res["scaling"] = scal
res["ratio_resid4_s1_over_s0.5"] = scal[0]["max|M-M2-M3|"] / scal[1]["max|M-M2-M3|"]
# non-product form: M3/M2 as a function of d (exact odd part / closed M2)
Me = np.array([mutual(L, tf, g1, g2, 0, d) for d in ds])
Mm = np.array([mutual(L, tf, -g1, -g2, 0, d) for d in ds])
m2 = m2_closed(L, tf, g1, g2, ds, n)
res["d"] = ds.tolist()
res["M_exact"] = Me.tolist()
res["M2_closed"] = m2.tolist()
res["M3_closed"] = m3_closed(L, tf, g1, g2, ds, n).tolist()
res["odd_part_over_M2"] = ((Me - Mm) / 2 / m2).tolist()
# highest state: M_top(g1,g2) = -M_low(-g1,-g2) for even L
if L % 2 == 0:
Mt = np.array([mutual(L, tf, g1, g2, 0, d, top=True) for d in ds])
res["max_dev_top_map"] = float(np.max(np.abs(Mt + Mm)))
return res
def main():
tf = 1.0
results = [run(8, tf, 0.04, -0.07), run(13, tf, 0.05, -0.03), run(13, tf, 0.03, 0.06)]
(OUT / "held_pair.json").write_text(json.dumps(results, indent=1))
lines = []
for r in results:
lines.append(f"L={r['L']} t_f={r['t_f']} g1={r['g1']} g2={r['g2']} "
f"(Kato bound {r['kato_bound_|g1|+|g2|<']:.4f})")
for k in ["max_err_Lambda_closed", "max_err_Lambda2_closed", "max_dev_translation",
"max_dev_reflection", "max_dev_exchange", "max_|F1+F2|", "max_dev_top_map"]:
if k in r:
lines.append(f" {k}: {r[k]:.3e}")
for s in r["scaling"]:
lines.append(" scale {scale}: max|M|={a:.3e} max|M-M2|={b:.3e} "
"max|M-M2-M3|={c:.3e} max|odd-M3|={e:.3e}".format(
scale=s["scale"], a=s["max|M|"], b=s["max|M-M2|"],
c=s["max|M-M2-M3|"], e=s["max|odd part - M3|"]))
lines.append(f" residual ratio (scale 1 / scale 1/2): {r['ratio_resid4_s1_over_s0.5']:.2f}"
" (16 expected for O(g^4))")
lines.append(" d, M_exact, M2, M3, (odd part)/M2:")
for d in r["d"]:
lines.append(f" {d:2d} {r['M_exact'][d]: .6e} {r['M2_closed'][d]: .6e} "
f"{r['M3_closed'][d]: .6e} {r['odd_part_over_M2'][d]: .5f}")
(OUT / "held_pair.log").write_text("\n".join(lines) + "\n")
print("\n".join(lines))
if __name__ == "__main__":
main()
check_mutual_fall.py23 KB · raw
"""Check of item 3 (mutual fall) of 07-ilang-weight/02-two-free-bodies, v2.
Two Model D bodies of different types (each in one clock level: half-gap M_i, motion amplitude tau_i,
branch sigma_i, coupling (a) or (c) with constant gamma_i) and N_f = 1 field object on a ring of L places.
The total contract commutes with the joint translation (2.1); it is diagonalized in the L sectors of the
total plane-wave label P (representative: field object at y = 0, body positions relative to it).
Start (R3): product of real Gaussian envelopes phi_i centred at c_i, in the adiabatic basis
|x1,sigma1,x2,sigma2> (x) Omega(x1,x2) (Omega the lowest or highest field state of the held problem),
projected on the band (the levels connected to that group) and normalized.
Observables (Step 5, (2.18)):
chart mean position <xhat_i>, xhat_i = sum_x chi_i(x) P_x^(i), chi_i(x) the representative of x in the
window {c_i - L//2, ..., c_i - L//2 + L - 1}; the cut bond joins the last and the first place of it;
a_i = d^2<xhat_i>/dlambda^2 at 0 = -<[C,[C,xhat_i]]>, exact (full-space reconstruction of the state);
a^w_i = <i[C,v_i]>, v_i the winding current; eps_i = a_i - a^w_i (cut term, (2.18));
a_evol = second derivative at 0 of <xhat_i>(lambda) from the joint evolution on [0, 60] (polynomial fit);
p_i^cut(lambda): weight of body i on the four places around the cut, at 0 and max over the run;
prediction (2.12): -E''_i sum (dchi_i(j)/2) w_i(j,l) d_i W(j,l) - (L/64) E''_i^2 Xi_i, E''_i from the band
of the single body with the field, dchi_i(j) = chi_i(j+1) - chi_i(j-1), Xi_i the cut sum of (2.12);
winding prediction (dchi = 2, no Xi term) for a^w_i;
exact zeroth-order part a_i^(0) (same start, the other body uncoupled) and the ratio of the mutual parts
(a_1 - a_1^(0)) / (a_2 - a_2^(0)) against -E''_1/E''_2 (2.13).
Isolation (R1): tau = 0 distances of the group to the levels with one branch flipped (delta_1), with the
same branches and an excited field (delta_f), with both branches flipped (delta_2), delta = min of these;
the same for each body alone; exact gap of the band at tau != 0 and weight of the band outside the group;
tau'_i/delta_1 and tau'^2/(delta_1 delta), tau' = tau (a), tau (1 + |gamma| N_f) (c).
Conserved operator (2.17): ||[Q_i, C]|| with Q_i = (sum_x (-1)^x P_x^(i)) sigma_3^(i), on body i and the
field object, for L = 14 (even) and L = 13 (odd, control).
Output: output/mutual_fall.json and output/mutual_fall.log
"""
import json
from pathlib import Path
import numpy as np
OUT = Path(__file__).parent / "output"
OUT.mkdir(exist_ok=True)
L = 14
TF = 0.5
LAM_MAX = 60.0
S1 = np.array([[0, 1], [1, 0]], dtype=complex)
S3 = np.diag([1.0, -1.0]).astype(complex)
I2 = np.eye(2)
IDX = {+1: 0, -1: 1}
PPOLY = np.polynomial.polynomial
def shift(n, k):
"""|x> -> |x+k> on a ring of n places."""
S = np.zeros((n, n))
for x in range(n):
S[(x + k) % n, x] = 1.0
return S
def h_field(pts, n=L):
H = -TF * (shift(n, 1) + shift(n, -1))
for x, g in pts:
H[x, x] += g
return H
def field_state(pts, top):
w, V = np.linalg.eigh(h_field(pts))
if top:
v, e = V[:, -1], w[-1]
v = v * np.sign(np.sum(v * (-1.0) ** np.arange(L)))
else:
v, e = V[:, 0], w[0]
v = v * np.sign(np.sum(v))
return e, v
def body_ops(b):
"""Own term + coupling (field object at y = 0) and winding current of one body, on (x, s): 2x+s."""
M, tau, gam, cpl = b["M"], b["tau"], b["gamma"], b["coupling"]
P0 = np.zeros((L, L)); P0[0, 0] = 1.0
H = np.kron(np.eye(L), M * S3) + gam * M * np.kron(P0, S3)
V = np.zeros((2 * L, 2 * L), dtype=complex)
for x in range(L):
f = 1.0
if cpl == "c":
f += gam * 0.5 * ((x == 0) + ((x + 1) % L == 0))
E = np.zeros((L, L)); E[(x + 1) % L, x] = 1.0 # |x+1><x|
H = H + 0.5 * tau * f * (np.kron(E, -1j * S1) + np.kron(E.T, 1j * S1))
V = V - 0.5 * tau * f * (np.kron(E, S1) + np.kron(E.T, S1))
return H, V
def body_field_ops(b, n):
"""Terms of C that act on body i, on (x, s) (x) field place y, ring of n places (N_f = 1).
Returns C_loc and Q_i = Pi sigma_3."""
M, tau, gam, cpl = b["M"], b["tau"], b["gamma"], b["coupling"]
Ix = np.eye(n)
def op(Ax, As, Ay):
return np.kron(np.kron(Ax, As), Ay)
C = op(Ix, I2, h_field([], n)).astype(complex)
for x in range(n):
Px = np.zeros((n, n)); Px[x, x] = 1.0
C = C + M * op(Px, S3, Ix) + gam * M * op(Px, S3, Px)
for x in range(n):
E = np.zeros((n, n)); E[(x + 1) % n, x] = 1.0
fac = Ix.copy()
if cpl == "c":
Ph = np.zeros((n, n)); Ph[x, x] += 0.5; Ph[(x + 1) % n, (x + 1) % n] += 0.5
fac = fac + gam * Ph
F = 0.5 * tau * op(E, -1j * S1, fac)
C = C + F + F.conj().T
Q = op(np.diag((-1.0) ** np.arange(n)), S3, Ix)
return C, Q
def chart(c):
"""Chart values chi(x) in the window {c - L//2, ..., c - L//2 + L - 1}; cut bond (top, top+1)."""
lo = c - L // 2
x = np.arange(L)
chi = (lo + (x - lo) % L).astype(float)
top = (lo + L - 1) % L
near = [(top + k) % L for k in (-1, 0, 1, 2)]
return chi, top, near
def tau_prime(b):
return b["tau"] * (1 + abs(b["gamma"])) if b["coupling"] == "c" else b["tau"]
def tau0_gaps(b1, b2, top):
"""Distances of the tau = 0 group from the other tau = 0 levels (two bodies; each body alone)."""
sel = L - 1 if top else 0
g1 = b1["sigma"] * b1["gamma"] * b1["M"]
g2 = b2["sigma"] * b2["gamma"] * b2["M"]
a1, a2 = b1["sigma"] * b1["M"], b2["sigma"] * b2["M"]
def spec(pts):
return np.linalg.eigvalsh(h_field(pts))
EG, same, f1, f12 = [], [], [], []
for d in range(L):
e = spec([(0, g1), (d, g2)])
EG.append(a1 + a2 + e[sel])
same.extend(a1 + a2 + np.delete(e, sel))
f1.extend(-a1 + a2 + spec([(0, -g1), (d, g2)]))
f1.extend(a1 - a2 + spec([(0, g1), (d, -g2)]))
f12.extend(-a1 - a2 + spec([(0, -g1), (d, -g2)]))
EG = np.array(EG)
def dist(S):
return float(np.min(np.abs(EG[:, None] - np.array(S)[None, :])))
out = {"delta_1": dist(f1), "delta_f": dist(same), "delta_2": dist(f12)}
out["delta"] = min(out["delta_1"], out["delta_f"], out["delta_2"])
for i, (b, g) in enumerate(((b1, g1), (b2, g2))):
sM = b["sigma"] * b["M"]
e, em = spec([(0, g)]), spec([(0, -g)])
eg = sM + e[sel]
out[f"body{i + 1}"] = {"delta_1": float(np.min(np.abs(eg - (-sM + em)))),
"delta_f": float(np.min(np.abs(eg - (sM + np.delete(e, sel)))))}
return out
def single_band_curv(b, top):
"""E'' at the rest point of the band of one body with the field (N_f = 1); band leak and exact gap."""
Hb, _ = body_ops(b)
Sm = np.kron(shift(L, -1), I2)
Sp = np.kron(shift(L, 1), I2)
g = b["sigma"] * b["gamma"] * b["M"]
_, om = field_state([(0, g)], top)
Es, leak, gap = [], 0.0, np.inf
for m in range(L):
P = 2 * np.pi * m / L
H = Hb - TF * (np.exp(-1j * P) * Sm + np.exp(1j * P) * Sp)
w, V = np.linalg.eigh(H)
G = np.zeros(2 * L, dtype=complex)
for t in range(L):
G[2 * ((-t) % L) + IDX[b["sigma"]]] = np.exp(-1j * P * t) * om[t]
ov = np.abs(G.conj() @ V) ** 2
k = int(np.argmax(ov))
assert ov[k] > 0.5
leak = max(leak, 1 - ov[k])
gap = min(gap, float(np.min(np.abs(np.delete(w, k) - w[k]))))
Es.append(w[k])
Es = np.array(Es)
rs = np.arange(-(L // 2) + 1, L // 2 + 1)
Ps = 2 * np.pi * np.arange(L) / L
kap = np.array([np.mean(Es * np.exp(1j * Ps * r)) for r in rs])
Prest = np.pi if top else 0.0
return float(np.real(-np.sum(rs ** 2 * kap * np.exp(-1j * Prest * rs)))), leak, gap
def gauss(center, w):
j = np.arange(L)
dj = (j - center + L / 2) % L - L / 2
f = np.exp(-dj ** 2 / (4 * w ** 2))
return f / np.linalg.norm(f)
def run_case(b1, b2, top, xc, widths, scale, only_acc=False):
b1 = dict(b1, tau=b1["tau"] * scale)
b2 = dict(b2, tau=b2["tau"] * scale)
if not only_acc:
# exact zeroth-order parts a_i^(0): the same start with the other body uncoupled
a01 = run_case(b1, dict(b2, gamma=0.0), top, xc, widths, 1.0, only_acc=True)[0]
a02 = run_case(dict(b1, gamma=0.0), b2, top, xc, widths, 1.0, only_acc=True)[1]
g1 = b1["sigma"] * b1["gamma"] * b1["M"]
g2 = b2["sigma"] * b2["gamma"] * b2["M"]
D = 4 * L * L
H1, V1 = body_ops(b1)
H2, V2 = body_ops(b2)
I = np.eye(2 * L)
H0 = np.kron(H1, I) + np.kron(I, H2)
v1 = np.kron(V1, I)
v2 = np.kron(I, V2)
Sm = np.kron(np.kron(shift(L, -1), I2), np.kron(shift(L, -1), I2))
Sp = Sm.T
# held problem: field states and eigenvalues W(j,l)
W = np.zeros((L, L))
Om = np.zeros((L, L, L))
for j in range(L):
for l in range(L):
W[j, l], Om[j, l] = field_state([(j, g1), (l, g2)], top)
phi1 = gauss(xc[0], widths[0])
phi2 = gauss(xc[1], widths[1])
s1, s2 = IDX[b1["sigma"]], IDX[b2["sigma"]]
def rep(x1, x2):
return ((2 * (x1 % L) + s1) * L + (x2 % L)) * 2 + s2
tt = np.arange(L)
X1 = (tt[None, :, None] + tt[:, None, None]) % L # [t, x1, 1] -> x1 + t
X2 = (tt[None, None, :] + tt[:, None, None]) % L # [t, 1, x2] -> x2 + t
F = phi1[X1] * phi2[X2] * Om[X1, X2, tt[:, None, None]]
norm2, store, leak, bgap = 0.0, [], 0.0, np.inf
for m in range(L):
P = 2 * np.pi * m / L
H = H0 - TF * (np.exp(-1j * P) * Sm + np.exp(1j * P) * Sp)
w, V = np.linalg.eigh(H)
G = np.zeros((D, L), dtype=complex)
for d in range(L):
for t in range(L):
G[rep(-t, d - t), d] = np.exp(-1j * P * t) * Om[0, d % L][t]
ov = np.sum(np.abs(G.conj().T @ V) ** 2, axis=0)
band = ov > 0.5
assert band.sum() == L, band.sum()
leak = max(leak, float(1 - ov[band].min()))
bgap = min(bgap, float(np.min(np.abs(w[band][:, None] - w[~band][None, :]))))
Vb, Eb = V[:, band], w[band]
psi = np.zeros(D, dtype=complex)
amp = np.einsum("t,txy->xy", np.exp(-1j * P * tt), F) / np.sqrt(L)
for x1 in range(L):
for x2 in range(L):
psi[rep(x1, x2)] = amp[x1, x2]
c = Vb.conj().T @ psi
norm2 += np.vdot(c, c).real
store.append((H, Vb, Eb, c))
store = [(H, Vb, Eb, c / np.sqrt(norm2)) for H, Vb, Eb, c in store]
Ebar = float(np.mean(np.concatenate([s[2] for s in store])))
# winding-current accelerations a^w_i = <i[C, v_i]>, sector by sector
aw = np.zeros(2)
for H, Vb, Eb, c in store:
ps = Vb @ c
for i, v in enumerate((v1, v2)):
aw[i] += np.real(np.vdot(ps, 1j * (H @ (v @ ps) - v @ (H @ ps))))
def full(power, lam=0.0):
"""(C - Ebar)^power exp(-i C lam) Psi(0) on the full space: [t, r1, s1, r2, s2] with
body 1 at r1 + t, body 2 at r2 + t, field object at t."""
vecs = np.array([Vb @ (((Eb - Ebar) ** power) * np.exp(-1j * Eb * lam) * c)
for _, Vb, Eb, c in store])
return (np.fft.ifft(vecs, axis=0) * np.sqrt(L)).reshape(L, L, 2, L, 2)
chi1, cut1, near1 = chart(xc[0])
chi2, cut2, near2 = chart(xc[1])
pos = (tt[:, None] + tt[None, :]) % L # pos[t, r] = r + t
CH = [chi1[pos][:, :, None, None, None], chi2[pos][:, None, None, :, None]]
NE = [np.isin(pos, near1)[:, :, None, None, None].astype(float),
np.isin(pos, near2)[:, None, None, :, None].astype(float)]
Ps0, CPs, C2Ps = full(0), full(1), full(2)
ach = np.array([2 * np.sum(CH[i] * np.abs(CPs) ** 2)
- 2 * np.real(np.sum(CH[i] * np.conj(C2Ps) * Ps0)) for i in range(2)])
vch = np.array([-2 * np.imag(np.sum(CH[i] * np.conj(CPs) * Ps0)) for i in range(2)])
if only_acc:
return ach
pcut0 = np.array([np.sum(NE[i] * np.abs(Ps0) ** 2) for i in range(2)])
# joint evolution of the chart mean positions
lam = np.linspace(0.0, LAM_MAX, 31)
xs = np.zeros((len(lam), 2))
pc = np.zeros((len(lam), 2))
for k, la in enumerate(lam):
pr = np.abs(full(0, la)) ** 2
xs[k] = [np.sum(CH[i] * pr) for i in range(2)]
pc[k] = [np.sum(NE[i] * pr) for i in range(2)]
u = lam / LAM_MAX
aev = np.array([2 * PPOLY.polyfit(u, xs[:, i], 6)[2] / LAM_MAX ** 2 for i in range(2)])
eps = ach - aw
# prediction (2.12): chart weights dchi/2 and the free-packet cut term -(L/64) E''^2 Xi
Ec1, leak1, gap1 = single_band_curv(b1, top)
Ec2, leak2, gap2 = single_band_curv(b2, top)
j = np.arange(L)
d1W = (W[(j + 1) % L, :] - W[(j - 1) % L, :]) / 2 # d1W[j, l]
d2W = (W[:, (j + 1) % L] - W[:, (j - 1) % L]) / 2 # d2W[j, l]
dchi1 = (chi1[(j + 1) % L] - chi1[(j - 1) % L]) / 2
dchi2 = (chi2[(j + 1) % L] - chi2[(j - 1) % L]) / 2
w1 = np.outer(phi1[(j + 1) % L] * phi1[(j - 1) % L] * dchi1, phi2 ** 2)
w2 = np.outer(phi1 ** 2, phi2[(j + 1) % L] * phi2[(j - 1) % L] * dchi2)
w1w = np.outer(phi1[(j + 1) % L] * phi1[(j - 1) % L], phi2 ** 2)
w2w = np.outer(phi1 ** 2, phi2[(j + 1) % L] * phi2[(j - 1) % L])
def xi(phi, a):
f = lambda k: phi[(a + k) % L]
return 2 * (f(-1) * f(3) + f(0) * f(4) - f(-3) * f(1) - f(-2) * f(2)
+ f(-1) ** 2 + f(0) ** 2 - f(1) ** 2 - f(2) ** 2)
a0 = np.array([-L / 64 * Ec1 ** 2 * xi(phi1, cut1), -L / 64 * Ec2 ** 2 * xi(phi2, cut2)])
pred1 = np.array([-Ec1 * np.sum(w1 * d1W), -Ec2 * np.sum(w2 * d2W)])
pred = pred1 + a0
predw = np.array([-Ec1 * np.sum(w1w * d1W), -Ec2 * np.sum(w2w * d2W)])
# in-situ estimate (2.16): relative change -/+ gamma_i * [1,2,1]/4-average of the change of the
# field density made by the other body at the hopping body's place; upper sign (a), lower (c)
drho2 = np.array([field_state([(l, g2)], top)[1] ** 2 - 1.0 / L for l in range(L)]) # [l, y]
drho1 = np.array([field_state([(l, g1)], top)[1] ** 2 - 1.0 / L for l in range(L)]) # [j, y]
av2 = (np.roll(drho2, 1, axis=1) + 2 * drho2 + np.roll(drho2, -1, axis=1)) / 4
av1 = (np.roll(drho1, 1, axis=1) + 2 * drho1 + np.roll(drho1, -1, axis=1)) / 4
R1 = av2.T # R1[j, l]: average density change by body 2 (at l) around place j
R2 = av1 # R2[j, l]: average density change by body 1 (at j) around place l
sg = [(-1.0 if b["coupling"] == "a" else 1.0) * b["gamma"] for b in (b1, b2)]
insitu = np.array([sg[0] * np.sum(w1 * R1 * d1W) / np.sum(w1 * d1W),
sg[1] * np.sum(w2 * R2 * d2W) / np.sum(w2 * d2W)])
tp = [tau_prime(b1), tau_prime(b2)]
return {"scale": scale, "a_chart": ach.tolist(), "a_winding": aw.tolist(), "eps_cut": eps.tolist(),
"pcut_0": pcut0.tolist(), "pcut_max_run": pc.max(axis=0).tolist(),
"a_evolution": aev.tolist(), "v_chart_0": vch.tolist(),
"a_pred": pred.tolist(), "a_pred_first_order": pred1.tolist(), "a_free_cut": a0.tolist(),
"a_pred_winding": predw.tolist(), "Epp": [Ec1, Ec2],
"a_zeroth_exact": [a01, a02],
"ratio_mutual": (ach[0] - a01) / (ach[1] - a02),
"ratio_mutual_over_inertia_minus_1": (ach[0] - a01) / (ach[1] - a02) / (-Ec1 / Ec2) - 1,
"ratio_first_order_pred_over_inertia_minus_1": pred1[0] / pred1[1] / (-Ec1 / Ec2) - 1,
"dev": (ach / pred - 1).tolist(), "dev_winding": (aw / predw - 1).tolist(),
"evol_vs_exact": (aev / ach - 1).tolist(),
"ratio_exact": ach[0] / ach[1], "ratio_pred": pred[0] / pred[1],
"inertia_ratio": -Ec1 / Ec2, "insitu_est": insitu.tolist(),
"band_gap_exact": bgap, "band_leak": leak,
"single_body": [{"leak": leak1, "gap_exact": gap1}, {"leak": leak2, "gap_exact": gap2}],
"tau_prime": tp,
"c_factors": [float(np.sum(phi1[(j + 1) % L] * phi1[(j - 1) % L])),
float(np.sum(phi2[(j + 1) % L] * phi2[(j - 1) % L]))]}
CASES = [
{"name": "A: (a), sigma=(+,+), lowest", "top": False,
"b1": {"M": 2.0, "tau": 0.10, "gamma": 0.025, "sigma": +1, "coupling": "a"},
"b2": {"M": 3.0, "tau": 0.15, "gamma": 0.02, "sigma": +1, "coupling": "a"},
"xc": (3, 8), "widths": (1.3, 1.3)},
{"name": "B: (c), sigma=(+,-), lowest", "top": False,
"b1": {"M": 2.5, "tau": 0.12, "gamma": 0.02, "sigma": +1, "coupling": "c"},
"b2": {"M": 1.6, "tau": 0.08, "gamma": 0.03, "sigma": -1, "coupling": "c"},
"xc": (3, 8), "widths": (1.3, 1.3)},
{"name": "C: (a), sigma=(-,-), highest", "top": True,
"b1": {"M": 2.0, "tau": 0.10, "gamma": 0.025, "sigma": -1, "coupling": "a"},
"b2": {"M": 3.0, "tau": 0.15, "gamma": 0.02, "sigma": -1, "coupling": "a"},
"xc": (3, 8), "widths": (1.3, 1.3)},
{"name": "D: (c), sigma=(+,+), lowest, unequal widths", "top": False,
"b1": {"M": 2.0, "tau": 0.10, "gamma": 0.025, "sigma": +1, "coupling": "c"},
"b2": {"M": 3.0, "tau": 0.15, "gamma": 0.02, "sigma": +1, "coupling": "c"},
"xc": (3, 8), "widths": (1.1, 1.6)},
{"name": "E: (c), sigma=(+,+), lowest", "top": False,
"b1": {"M": 2.0, "tau": 0.10, "gamma": 0.025, "sigma": +1, "coupling": "c"},
"b2": {"M": 3.0, "tau": 0.15, "gamma": 0.02, "sigma": +1, "coupling": "c"},
"xc": (3, 8), "widths": (1.3, 1.3)},
]
GAMMA_SCAN = []
for base in (CASES[0], CASES[4]):
GAMMA_SCAN.append({"name": base["name"][:2] + " with gamma_1, gamma_2 times 0.5", "top": False,
"b1": dict(base["b1"], gamma=base["b1"]["gamma"] * 0.5),
"b2": dict(base["b2"], gamma=base["b2"]["gamma"] * 0.5),
"xc": (3, 8), "widths": (1.3, 1.3)})
def q_check():
"""||[Q_i, C_loc]|| for the bodies of cases A (coupling a) and E (coupling c), L = 14 and L = 13."""
res = []
for case in (CASES[0], CASES[4]):
for b in (case["b1"], case["b2"]):
for n in (14, 13):
Cl, Q = body_field_ops(b, n)
res.append({"case": case["name"][:2], "coupling": b["coupling"], "M": b["M"], "L": n,
"norm_comm": float(np.linalg.norm(Q @ Cl - Cl @ Q, 2))})
return res
def main():
results = {"L": L, "t_f": TF, "N_f": 1, "lambda_max": LAM_MAX, "q_check": q_check(), "cases": []}
lines = [f"L={L} t_f={TF} N_f=1; scales s multiply tau_1, tau_2; evolution on [0, {LAM_MAX}]"]
for q in results["q_check"]:
lines.append(f" (2.17) ||[Q,C]||: case {q['case']} coupling {q['coupling']} M={q['M']} "
f"L={q['L']}: {q['norm_comm']:.2e}")
for case in CASES + GAMMA_SCAN:
gaps = tau0_gaps(case["b1"], case["b2"], case["top"])
runs = [run_case(case["b1"], case["b2"], case["top"], case["xc"], case["widths"], s)
for s in (1.0, 0.5, 0.25)]
ss = np.array([r["scale"] for r in runs])
A = np.vstack([np.ones(3), ss ** 2, ss ** 4]).T
dev0 = [float(np.linalg.solve(A, np.array([r["dev"][i] for r in runs]))[0]) for i in range(2)]
rr = np.array([r["ratio_exact"] / r["ratio_pred"] - 1 for r in runs])
rdev0 = float(np.linalg.solve(A, rr)[0])
for r in runs:
r["tau_prime_over_delta_1"] = max(r["tau_prime"]) / gaps["delta_1"]
r["tau_prime2_over_delta_1_delta"] = max(r["tau_prime"]) ** 2 / (gaps["delta_1"] * gaps["delta"])
res = {"case": case["name"], "params": {k: case[k] for k in ("b1", "b2", "xc", "widths", "top")},
"tau0_gaps": gaps, "runs": runs, "dev_extrapolated_tau0": dev0,
"ratio_dev_extrapolated_tau0": rdev0, "insitu_estimate": runs[0]["insitu_est"],
"insitu_ratio_estimate": runs[0]["insitu_est"][0] - runs[0]["insitu_est"][1]}
results["cases"].append(res)
n0 = len(lines)
lines.append(case["name"] + f" b1={case['b1']} b2={case['b2']} centres={case['xc']}"
f" widths={case['widths']}")
lines.append(" tau=0 gaps: delta_1={:.4f} delta_f={:.4f} delta_2={:.4f} delta={:.4f}; body1 "
"(delta_1={:.4f}, delta_f={:.4f}) body2 (delta_1={:.4f}, delta_f={:.4f})".format(
gaps["delta_1"], gaps["delta_f"], gaps["delta_2"], gaps["delta"],
gaps["body1"]["delta_1"], gaps["body1"]["delta_f"],
gaps["body2"]["delta_1"], gaps["body2"]["delta_f"]))
for r in runs:
lines.append(" s={}: a=({:.6e},{:.6e}) a_evol/a-1=({:.1e},{:.1e}) v(0)=({:.1e},{:.1e})".format(
r["scale"], *r["a_chart"], *r["evol_vs_exact"], *r["v_chart_0"]))
lines.append(" a^w=({:.6e},{:.6e}) eps_cut=a-a^w=({:.2e},{:.2e}) "
"p_cut(0)=({:.1e},{:.1e}) max p_cut=({:.1e},{:.1e})".format(
*r["a_winding"], *r["eps_cut"], *r["pcut_0"], *r["pcut_max_run"]))
lines.append(" a_pred=({:.6e},{:.6e}) [free-packet term ({:.2e},{:.2e}); exact a^(0)=({:.2e},{:.2e})]"
" E''=({:.6e},{:.6e}) dev=({:.3e},{:.3e}) dev_w=({:.3e},{:.3e})".format(
*r["a_pred"], *r["a_free_cut"], *r["a_zeroth_exact"], *r["Epp"], *r["dev"],
*r["dev_winding"]))
lines.append(" ratio exact={:.6f} pred={:.6f} -E''1/E''2={:.6f}; mutual ratio (a-a^(0))={:.6f}, "
"/(-E''1/E''2)-1={:.2e} (first-order pred: {:.2e}) c=({:.4f},{:.4f})".format(
r["ratio_exact"], r["ratio_pred"], r["inertia_ratio"], r["ratio_mutual"],
r["ratio_mutual_over_inertia_minus_1"],
r["ratio_first_order_pred_over_inertia_minus_1"], *r["c_factors"]))
lines.append(" band gap={:.4f} band leak={:.1e} single (gap, leak)=({:.4f},{:.1e}),"
"({:.4f},{:.1e}) tau'/delta_1={:.3f} tau'^2/(delta_1 delta)={:.3f}".format(
r["band_gap_exact"], r["band_leak"], r["single_body"][0]["gap_exact"],
r["single_body"][0]["leak"], r["single_body"][1]["gap_exact"],
r["single_body"][1]["leak"], r["tau_prime_over_delta_1"],
r["tau_prime2_over_delta_1_delta"]))
lines.append(" extrapolated (s->0) dev_i = ({:.3e}, {:.3e}); in-situ estimate (2.16) = ({:.3e}, {:.3e})"
.format(dev0[0], dev0[1], res["insitu_estimate"][0], res["insitu_estimate"][1]))
lines.append(" extrapolated ratio deviation = {:.3e}; in-situ estimate (2.19) = {:.3e}"
.format(rdev0, res["insitu_ratio_estimate"]))
print("\n".join(lines[n0:]), flush=True)
(OUT / "mutual_fall.json").write_text(json.dumps(results, indent=1))
(OUT / "mutual_fall.log").write_text("\n".join(lines) + "\n")
if __name__ == "__main__":
main()
output/held_pair.json7 KB · raw
[
{
"L": 8,
"t_f": 1.0,
"g1": 0.04,
"g2": -0.07,
"kato_bound_|g1|+|g2|<": 0.2928932188134525,
"max_err_Lambda_closed": 4.996003610813204e-16,
"max_err_Lambda2_closed": 8.881784197001252e-16,
"max_dev_translation": 3.1086244689504383e-15,
"max_dev_reflection": 2.6645352591003757e-15,
"max_dev_exchange": 1.9984014443252818e-15,
"max_|F1+F2|": 2.886579864025407e-15,
"scaling": [
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"max|odd part - M3|": 1.625444594766722e-08
},
{
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"max|M-M2|": 1.3122086886351427e-06,
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"max|odd part - M3|": 5.088112074177463e-10
},
{
"scale": 0.25,
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"max|M-M2|": 1.6276485860053678e-07,
"max|M-M2-M3|": 1.3056856839334916e-09,
"max|odd part - M3|": 1.5907177356680298e-11
}
],
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],
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1.1484375000000004e-06
],
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},
{
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"g1": 0.05,
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"max_dev_exchange": 2.220446049250313e-15,
"max_|F1+F2|": 9.992007221626409e-16,
"scaling": [
{
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"max|M-M2-M3|": 2.7642075433761645e-07,
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},
{
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},
{
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}
],
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1.2977696859353665e-05
],
"odd_part_over_M2": [
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}
]output/held_pair.log4 KB · raw
L=8 t_f=1.0 g1=0.04 g2=-0.07 (Kato bound 0.2929)
max_err_Lambda_closed: 4.996e-16
max_err_Lambda2_closed: 8.882e-16
max_dev_translation: 3.109e-15
max_dev_reflection: 2.665e-15
max_dev_exchange: 1.998e-15
max_|F1+F2|: 2.887e-15
max_dev_top_map: 1.554e-15
scale 1.0: max|M|=4.700e-04 max|M-M2|=1.066e-05 max|M-M2-M3|=3.465e-07 max|odd-M3|=1.625e-08
scale 0.5: max|M|=1.162e-04 max|M-M2|=1.312e-06 max|M-M2-M3|=2.115e-08 max|odd-M3|=5.088e-10
scale 0.25: max|M|=2.887e-05 max|M-M2|=1.628e-07 max|M-M2-M3|=1.306e-09 max|odd-M3|=1.591e-11
residual ratio (scale 1 / scale 1/2): 16.39 (16 expected for O(g^4))
d, M_exact, M2, M3, (odd part)/M2:
0 4.700330e-04 4.593750e-04 1.033594e-05 0.02250
1 1.541747e-04 1.531250e-04 1.148438e-06 0.00745
2 -6.789250e-05 -6.562500e-05 -2.132813e-06 0.03249
3 -1.994943e-04 -1.968750e-04 -2.789063e-06 0.01411
4 -2.430676e-04 -2.406250e-04 -2.789063e-06 0.01152
5 -1.994943e-04 -1.968750e-04 -2.789063e-06 0.01411
6 -6.789250e-05 -6.562500e-05 -2.132813e-06 0.03249
7 1.541747e-04 1.531250e-04 1.148438e-06 0.00745
L=13 t_f=1.0 g1=0.05 g2=-0.03 (Kato bound 0.1145)
max_err_Lambda_closed: 7.772e-16
max_err_Lambda2_closed: 5.773e-15
max_dev_translation: 3.109e-15
max_dev_reflection: 1.332e-15
max_dev_exchange: 2.220e-15
max_|F1+F2|: 9.992e-16
scale 1.0: max|M|=2.425e-04 max|M-M2|=6.032e-06 max|M-M2-M3|=2.764e-07 max|odd-M3|=1.176e-08
scale 0.5: max|M|=6.136e-05 max|M-M2|=7.712e-07 max|M-M2-M3|=1.733e-08 max|odd-M3|=3.683e-10
scale 0.25: max|M|=1.544e-05 max|M-M2|=9.749e-08 max|M-M2-M3|=1.084e-09 max|odd-M3|=1.152e-11
residual ratio (scale 1 / scale 1/2): 15.95 (16 expected for O(g^4))
d, M_exact, M2, M3, (odd part)/M2:
0 2.424885e-04 2.485207e-04 -6.308603e-06 -0.02539
1 1.396928e-04 1.420118e-04 -2.403277e-06 -0.01689
2 5.302272e-05 5.325444e-05 -1.502048e-07 -0.00271
3 -1.686655e-05 -1.775148e-05 9.968138e-07 -0.05625
4 -6.955535e-05 -7.100592e-05 1.474738e-06 -0.02071
5 -1.047922e-04 -1.065089e-04 1.611288e-06 -0.01504
6 -1.224402e-04 -1.242604e-04 1.624943e-06 -0.01298
7 -1.224402e-04 -1.242604e-04 1.624943e-06 -0.01298
8 -1.047922e-04 -1.065089e-04 1.611288e-06 -0.01504
9 -6.955535e-05 -7.100592e-05 1.474738e-06 -0.02071
10 -1.686655e-05 -1.775148e-05 9.968138e-07 -0.05625
11 5.302272e-05 5.325444e-05 -1.502048e-07 -0.00271
12 1.396928e-04 1.420118e-04 -2.403277e-06 -0.01689
L=13 t_f=1.0 g1=0.03 g2=0.06 (Kato bound 0.1145)
max_err_Lambda_closed: 7.772e-16
max_err_Lambda2_closed: 5.773e-15
max_dev_translation: 3.553e-15
max_dev_reflection: 8.882e-16
max_dev_exchange: 3.553e-15
max_|F1+F2|: 1.110e-15
scale 1.0: max|M|=2.662e-04 max|M-M2|=3.201e-05 max|M-M2-M3|=2.058e-06 max|odd-M3|=6.077e-08
scale 0.5: max|M|=7.043e-05 max|M-M2|=4.129e-06 max|M-M2-M3|=1.297e-07 max|odd-M3|=1.895e-09
scale 0.25: max|M|=1.811e-05 max|M-M2|=5.242e-07 max|M-M2-M3|=8.133e-09 max|odd-M3|=5.919e-11
residual ratio (scale 1 / scale 1/2): 15.87 (16 expected for O(g^4))
d, M_exact, M2, M3, (odd part)/M2:
0 -2.662168e-04 -2.982249e-04 3.406645e-05 -0.11429
1 -1.576592e-04 -1.704142e-04 1.297770e-05 -0.07580
2 -6.273501e-05 -6.390533e-05 8.111061e-07 -0.01208
3 1.633884e-05 2.130178e-05 -5.382795e-06 -0.25333
4 7.758364e-05 8.520710e-05 -7.963587e-06 -0.09347
5 1.193799e-04 1.278107e-04 -8.700956e-06 -0.06804
6 1.405770e-04 1.491124e-04 -8.774693e-06 -0.05880
7 1.405770e-04 1.491124e-04 -8.774693e-06 -0.05880
8 1.193799e-04 1.278107e-04 -8.700956e-06 -0.06804
9 7.758364e-05 8.520710e-05 -7.963587e-06 -0.09347
10 1.633884e-05 2.130178e-05 -5.382795e-06 -0.25333
11 -6.273501e-05 -6.390533e-05 8.111061e-07 -0.01208
12 -1.576592e-04 -1.704142e-04 1.297770e-05 -0.07580
output/mutual_fall.json59 KB · raw
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],
"ratio_mutual": -0.6666050877651133,
"ratio_mutual_over_inertia_minus_1": -4.4276942756660276e-05,
"ratio_first_order_pred_over_inertia_minus_1": -2.9952765209451115e-05,
"dev": [
2.988307322104866e-05,
4.524451867582435e-05
],
"dev_winding": [
3.0335889917409986e-05,
4.418331660049901e-05
],
"evol_vs_exact": [
-9.096120190399404e-09,
-3.0789537808217915e-08
],
"ratio_exact": -0.6630946969627064,
"ratio_pred": -0.6631048827513425,
"inertia_ratio": -0.6666346043073278,
"insitu_est": [
-1.157621890164934e-05,
-7.834989599516703e-06
],
"band_gap_exact": 0.09507257818617543,
"band_leak": 7.236927057285847e-05,
"single_body": [
{
"leak": 3.705829180478126e-05,
"gap_exact": 0.0973179058804885
},
{
"leak": 3.7069033748449165e-05,
"gap_exact": 0.09698969265840685
}
],
"tau_prime": [
0.025,
0.0375
],
"c_factors": [
0.7439002134928834,
0.7439002134928835
],
"tau_prime_over_delta_1": 0.0187244087338692,
"tau_prime2_over_delta_1_delta": 0.007373678638832345
}
],
"dev_extrapolated_tau0": [
-1.131252546904759e-05,
-8.144235799549017e-06
],
"ratio_dev_extrapolated_tau0": -3.1647686673667353e-06,
"insitu_estimate": [
-1.157621890164934e-05,
-7.834989599516703e-06
],
"insitu_ratio_estimate": -3.7412293021326374e-06
},
{
"case": "E: with gamma_1, gamma_2 times 0.5",
"params": {
"b1": {
"M": 2.0,
"tau": 0.1,
"gamma": 0.0125,
"sigma": 1,
"coupling": "c"
},
"b2": {
"M": 3.0,
"tau": 0.15,
"gamma": 0.01,
"sigma": 1,
"coupling": "c"
},
"xc": [
3,
8
],
"widths": [
1.3,
1.3
],
"top": false
},
"tau0_gaps": {
"delta_1": 2.002733465872758,
"delta_f": 0.09522591936977687,
"delta_2": 8.006948044096603,
"delta": 0.09522591936977687,
"body1": {
"delta_1": 2.003373339336669,
"delta_f": 0.09734446242924655
},
"body2": {
"delta_1": 4.004003009464755,
"delta_f": 0.09702962736520337
}
},
"runs": [
{
"scale": 1.0,
"a_chart": [
9.961720637902333e-08,
-1.6252818705416903e-07
],
"a_winding": [
1.0309902040229303e-07,
-1.54616743630061e-07
],
"eps_cut": [
-3.4818140232696937e-09,
-7.911443424108036e-09
],
"pcut_0": [
0.00020398701490375692,
0.00020398848580910507
],
"pcut_max_run": [
0.00023132542098986244,
0.00026542079380959796
],
"a_evolution": [
9.961720641876274e-08,
-1.6252818622800356e-07
],
"v_chart_0": [
1.5496997840765414e-15,
7.711991260953534e-16
],
"a_pred": [
9.956684691017993e-08,
-1.6238731000073707e-07
],
"a_pred_first_order": [
1.0305619657296214e-07,
-1.545411274738116e-07
],
"a_free_cut": [
-3.489349662782211e-09,
-7.846182526925485e-09
],
"a_pred_winding": [
1.0303572426381479e-07,
-1.5450579957065156e-07
],
"Epp": [
0.005002608108152657,
0.007501592008132361
],
"a_zeroth_exact": [
-3.5026058189767525e-09,
-7.875760085884601e-09
],
"ratio_mutual": -0.6667843125355384,
"ratio_mutual_over_inertia_minus_1": -0.0001327783575431063,
"ratio_first_order_pred_over_inertia_minus_1": -2.9952765209451115e-05,
"dev": [
0.0005057855140158107,
0.000867537330542012
],
"dev_winding": [
0.0006143125496567148,
0.0007180575727108174
],
"evol_vs_exact": [
3.9892111836081767e-10,
-5.083213516776652e-09
],
"ratio_exact": -0.6129226455090028,
"ratio_pred": -0.6131442593003603,
"inertia_ratio": -0.6668728588184224,
"insitu_est": [
1.157621890164934e-05,
7.834989599516703e-06
],
"band_gap_exact": 0.09092582345185374,
"band_leak": 0.001185294132223902,
"single_body": [
{
"leak": 0.0005929385633378637,
"gap_exact": 0.09686924755088566
},
{
"leak": 0.0005929583159981489,
"gap_exact": 0.09629948319981319
}
],
"tau_prime": [
0.10125,
0.1515
],
"c_factors": [
0.7439002134928834,
0.7439002134928835
],
"tau_prime_over_delta_1": 0.07564661128483156,
"tau_prime2_over_delta_1_delta": 0.120350233271566
},
{
"scale": 0.5,
"a_chart": [
2.554948486977869e-08,
-3.9133293052532707e-08
],
"a_winding": [
2.576264112913557e-08,
-3.863358177654601e-08
],
"eps_cut": [
-2.131562593568828e-10,
-4.997112759866955e-10
],
"pcut_0": [
0.00020319871060152733,
0.00020319878484904494
],
"pcut_max_run": [
0.00020496454979190945,
0.00020707962680198072
],
"a_evolution": [
2.554948495613656e-08,
-3.913329309320272e-08
],
"v_chart_0": [
7.090749299485515e-16,
-9.121443489169103e-17
],
"a_pred": [
2.554608812137846e-08,
-3.9126086582561974e-08
],
"a_pred_first_order": [
2.5764174599195793e-08,
-3.863568981858934e-08
],
"a_free_cut": [
-2.1808647781733437e-10,
-4.903967639726352e-10
],
"a_pred_winding": [
2.575905649698689e-08,
-3.862685774954246e-08
],
"Epp": [
0.0012506581169871422,
0.001875417804370069
],
"a_zeroth_exact": [
-2.182936561538474e-10,
-4.908588624098927e-10
],
"ratio_mutual": -0.6668259664790708,
"ratio_mutual_over_inertia_minus_1": -6.462760279268842e-05,
"ratio_first_order_pred_over_inertia_minus_1": -2.995276520911805e-05,
"dev": [
0.00013296550078778502,
0.0001841858105466887
],
"dev_winding": [
0.00013916007168579547,
0.0001740764689468488
],
"evol_vs_exact": [
3.380024038435181e-09,
1.0392688931659677e-09
],
"ratio_exact": -0.6528835903352408,
"ratio_pred": -0.6529170267890795,
"inertia_ratio": -0.6668690646280943,
"insitu_est": [
1.157621890164934e-05,
7.834989599516703e-06
],
"band_gap_exact": 0.09440436026693266,
"band_leak": 0.00029599323515483444,
"single_body": [
{
"leak": 0.00014842739772713553,
"gap_exact": 0.09723490900488962
},
{
"leak": 0.00014842190471264782,
"gap_exact": 0.09686431003094009
}
],
"tau_prime": [
0.050625,
0.07575
],
"c_factors": [
0.7439002134928834,
0.7439002134928835
],
"tau_prime_over_delta_1": 0.03782330564241578,
"tau_prime2_over_delta_1_delta": 0.0300875583178915
},
{
"scale": 0.25,
"a_chart": [
6.427681404085343e-09,
-9.690064565390518e-09
],
"a_winding": [
6.440034375101487e-09,
-9.657198094326671e-09
],
"eps_cut": [
-1.2352971016143467e-11,
-3.2866471063847225e-11
],
"pcut_0": [
0.00020300203570132143,
0.00020300203550342337
],
"pcut_max_run": [
0.00020312894622244053,
0.00020326464086177466
],
"a_evolution": [
6.427681515627411e-09,
-9.690064718143815e-09
],
"v_chart_0": [
-2.0380274964907284e-16,
4.160835443975635e-16
],
"a_pred": [
6.427421184648607e-09,
-9.689598546903644e-09
],
"a_pred_first_order": [
6.441051623258757e-09,
-9.658948583331481e-09
],
"a_free_cut": [
-1.3630438610149492e-11,
-3.0649963572163205e-11
],
"a_pred_winding": [
6.439772096122589e-09,
-9.656740560096776e-09
],
"Epp": [
0.0003126649162986661,
0.00046885571940685766
],
"a_zeroth_exact": [
-1.363367595072295e-11,
-3.065718377299145e-11
],
"ratio_mutual": -0.6668437126168115,
"ratio_mutual_over_inertia_minus_1": -3.6549312619871266e-05,
"ratio_first_order_pred_over_inertia_minus_1": -2.995276520922907e-05,
"dev": [
4.0485823047831104e-05,
4.809471564981571e-05
],
"dev_winding": [
4.072799083298939e-05,
4.7379778616685186e-05
],
"evol_vs_exact": [
1.7353390946439617e-08,
1.5763909066990323e-08
],
"ratio_exact": -0.6633269944395156,
"ratio_pred": -0.6633320414190451,
"inertia_ratio": -0.6668680861869698,
"insitu_est": [
1.157621890164934e-05,
7.834989599516703e-06
],
"band_gap_exact": 0.09507208552793678,
"band_leak": 7.248328425601969e-05,
"single_body": [
{
"leak": 3.711892408042594e-05,
"gap_exact": 0.09731766401074537
},
{
"leak": 3.711691526819827e-05,
"gap_exact": 0.096989406695005
}
],
"tau_prime": [
0.0253125,
0.037875
],
"c_factors": [
0.7439002134928834,
0.7439002134928835
],
"tau_prime_over_delta_1": 0.01891165282120789,
"tau_prime2_over_delta_1_delta": 0.007521889579472875
}
],
"dev_extrapolated_tau0": [
9.72373718491782e-06,
5.819620471031982e-06
],
"ratio_dev_extrapolated_tau0": 3.907413032945399e-06,
"insitu_estimate": [
1.157621890164934e-05,
7.834989599516703e-06
],
"insitu_ratio_estimate": 3.7412293021326374e-06
}
]
}output/mutual_fall.log20 KB · raw
L=14 t_f=0.5 N_f=1; scales s multiply tau_1, tau_2; evolution on [0, 60.0]
(2.17) ||[Q,C]||: case A: coupling a M=2.0 L=14: 0.00e+00
(2.17) ||[Q,C]||: case A: coupling a M=2.0 L=13: 1.00e-01
(2.17) ||[Q,C]||: case A: coupling a M=3.0 L=14: 0.00e+00
(2.17) ||[Q,C]||: case A: coupling a M=3.0 L=13: 1.50e-01
(2.17) ||[Q,C]||: case E: coupling c M=2.0 L=14: 0.00e+00
(2.17) ||[Q,C]||: case E: coupling c M=2.0 L=13: 1.01e-01
(2.17) ||[Q,C]||: case E: coupling c M=3.0 L=14: 0.00e+00
(2.17) ||[Q,C]||: case E: coupling c M=3.0 L=13: 1.51e-01
A: (a), sigma=(+,+), lowest b1={'M': 2.0, 'tau': 0.1, 'gamma': 0.025, 'sigma': 1, 'coupling': 'a'} b2={'M': 3.0, 'tau': 0.15, 'gamma': 0.02, 'sigma': 1, 'coupling': 'a'} centres=(3, 8) widths=(1.3, 1.3)
tau=0 gaps: delta_1=2.0039 delta_f=0.0917 delta_2=8.0124 delta=0.0917; body1 (delta_1=2.0064, delta_f=0.0958) body2 (delta_1=4.0075, delta_f=0.0953)
s=1.0: a=(3.785826e-07,-5.810415e-07) a_evol/a-1=(-1.1e-09,-3.7e-09) v(0)=(-5.1e-16,4.6e-16)
a^w=(3.819916e-07,-5.730847e-07) eps_cut=a-a^w=(-3.41e-09,-7.96e-09) p_cut(0)=(2.0e-04,2.0e-04) max p_cut=(2.3e-04,2.7e-04)
a_pred=(3.779570e-07,-5.798805e-07) [free-packet term (-3.47e-09,-7.80e-09); exact a^(0)=(-3.48e-09,-7.83e-09)] E''=(4.986555e-03,7.478886e-03) dev=(1.655e-03,2.002e-03) dev_w=(1.675e-03,1.972e-03)
ratio exact=-0.651559 pred=-0.651784 -E''1/E''2=-0.666751; mutual ratio (a-a^(0))=-0.666525, /(-E''1/E''2)-1=-3.38e-04 (first-order pred: -3.17e-05) c=(0.7439,0.7439)
band gap=0.0879 band leak=1.2e-03 single (gap, leak)=(0.0954,5.9e-04),(0.0946,5.9e-04) tau'/delta_1=0.075 tau'^2/(delta_1 delta)=0.123
s=0.5: a=(9.517570e-08,-1.435804e-07) a_evol/a-1=(7.1e-10,-5.4e-09) v(0)=(4.5e-16,-1.7e-15)
a^w=(9.537485e-08,-1.430610e-07) eps_cut=a-a^w=(-1.99e-10,-5.19e-10) p_cut(0)=(2.0e-04,2.0e-04) max p_cut=(2.1e-04,2.1e-04)
a_pred=(9.514152e-08,-1.435145e-07) [free-packet term (-2.17e-10,-4.87e-10); exact a^(0)=(-2.17e-10,-4.88e-10)] E''=(1.246668e-03,1.869808e-03) dev=(3.593e-04,4.594e-04) dev_w=(3.608e-04,4.557e-04)
ratio exact=-0.662874 pred=-0.662940 -E''1/E''2=-0.666736; mutual ratio (a-a^(0))=-0.666650, /(-E''1/E''2)-1=-1.29e-04 (first-order pred: -3.17e-05) c=(0.7439,0.7439)
band gap=0.0910 band leak=2.9e-04 single (gap, leak)=(0.0957,1.5e-04),(0.0951,1.5e-04) tau'/delta_1=0.037 tau'^2/(delta_1 delta)=0.031
s=0.25: a=(2.382741e-08,-3.579076e-08) a_evol/a-1=(5.0e-09,-6.5e-09) v(0)=(-3.3e-16,-8.2e-16)
a^w=(2.383652e-08,-3.575247e-08) eps_cut=a-a^w=(-9.11e-12,-3.83e-11) p_cut(0)=(2.0e-04,2.0e-04) max p_cut=(2.0e-04,2.0e-04)
a_pred=(2.382615e-08,-3.578765e-08) [free-packet term (-1.35e-11,-3.05e-11); exact a^(0)=(-1.35e-11,-3.05e-11)] E''=(3.116688e-04,4.674576e-04) dev=(5.297e-05,8.691e-05) dev_w=(5.310e-05,8.620e-05)
ratio exact=-0.665742 pred=-0.665765 -E''1/E''2=-0.666732; mutual ratio (a-a^(0))=-0.666688, /(-E''1/E''2)-1=-6.54e-05 (first-order pred: -3.17e-05) c=(0.7439,0.7439)
band gap=0.0916 band leak=6.7e-05 single (gap, leak)=(0.0958,3.7e-05),(0.0952,3.7e-05) tau'/delta_1=0.019 tau'^2/(delta_1 delta)=0.008
extrapolated (s->0) dev_i = (-4.756e-05, -3.607e-05); in-situ estimate (2.16) = (-4.794e-05, -3.287e-05)
extrapolated ratio deviation = -1.149e-05; in-situ estimate (2.19) = -1.507e-05
B: (c), sigma=(+,-), lowest b1={'M': 2.5, 'tau': 0.12, 'gamma': 0.02, 'sigma': 1, 'coupling': 'c'} b2={'M': 1.6, 'tau': 0.08, 'gamma': 0.03, 'sigma': -1, 'coupling': 'c'} centres=(3, 8) widths=(1.3, 1.3)
tau=0 gaps: delta_1=3.0046 delta_f=0.0920 delta_2=0.0939 delta=0.0920; body1 (delta_1=3.0064, delta_f=0.0958) body2 (delta_1=3.2069, delta_f=0.0959)
s=1.0: a=(-4.084886e-07,-2.829296e-07) a_evol/a-1=(-8.4e-10,3.4e-09) v(0)=(-4.3e-18,2.5e-17)
a^w=(-4.037584e-07,-2.806167e-07) eps_cut=a-a^w=(-4.73e-09,-2.31e-09) p_cut(0)=(2.0e-04,2.0e-04) max p_cut=(2.4e-04,2.2e-04)
a_pred=(-4.080273e-07,-2.826472e-07) [free-packet term (-4.63e-09,-2.24e-09); exact a^(0)=(-4.64e-09,-2.24e-09)] E''=(5.760574e-03,-4.004179e-03) dev=(1.131e-03,9.993e-04) dev_w=(1.103e-03,9.742e-04)
ratio exact=1.443782 pred=1.443592 -E''1/E''2=1.438641; mutual ratio (a-a^(0))=1.438786, /(-E''1/E''2)-1=1.01e-04 (first-order pred: -2.76e-05) c=(0.7439,0.7439)
band gap=0.0895 band leak=1.1e-03 single (gap, leak)=(0.0953,5.5e-04),(0.0956,5.9e-04) tau'/delta_1=0.041 tau'^2/(delta_1 delta)=0.054
s=0.5: a=(-1.011674e-07,-7.026011e-08) a_evol/a-1=(-5.7e-12,1.1e-08) v(0)=(1.2e-16,-9.3e-18)
a^w=(-1.008563e-07,-7.010320e-08) eps_cut=a-a^w=(-3.11e-10,-1.57e-10) p_cut(0)=(2.0e-04,2.0e-04) max p_cut=(2.1e-04,2.0e-04)
a_pred=(-1.011415e-07,-7.024128e-08) [free-packet term (-2.89e-10,-1.40e-10); exact a^(0)=(-2.89e-10,-1.40e-10)] E''=(1.440175e-03,-1.001025e-03) dev=(2.565e-04,2.679e-04) dev_w=(2.544e-04,2.661e-04)
ratio exact=1.439899 pred=1.439915 -E''1/E''2=1.438699; mutual ratio (a-a^(0))=1.438643, /(-E''1/E''2)-1=-3.93e-05 (first-order pred: -2.76e-05) c=(0.7439,0.7439)
band gap=0.0917 band leak=2.4e-04 single (gap, leak)=(0.0957,1.4e-04),(0.0958,1.5e-04) tau'/delta_1=0.020 tau'^2/(delta_1 delta)=0.014
s=0.25: a=(-2.523252e-08,-1.753578e-08) a_evol/a-1=(3.5e-09,-7.7e-09) v(0)=(-1.0e-16,4.1e-16)
a^w=(-2.520901e-08,-1.752279e-08) eps_cut=a-a^w=(-2.35e-11,-1.30e-11) p_cut(0)=(2.0e-04,2.0e-04) max p_cut=(2.0e-04,2.0e-04)
a_pred=(-2.523129e-08,-1.753404e-08) [free-packet term (-1.81e-11,-8.73e-12); exact a^(0)=(-1.81e-11,-8.73e-12)] E''=(3.600457e-04,-2.502552e-04) dev=(4.868e-05,9.922e-05) dev_w=(4.848e-05,9.904e-05)
ratio exact=1.438916 pred=1.438989 -E''1/E''2=1.438714; mutual ratio (a-a^(0))=1.438602, /(-E''1/E''2)-1=-7.82e-05 (first-order pred: -2.76e-05) c=(0.7439,0.7439)
band gap=0.0920 band leak=5.7e-05 single (gap, leak)=(0.0958,3.4e-05),(0.0959,3.7e-05) tau'/delta_1=0.010 tau'^2/(delta_1 delta)=0.003
extrapolated (s->0) dev_i = (-1.965e-05, 4.423e-05); in-situ estimate (2.16) = (-3.193e-05, 3.613e-05)
extrapolated ratio deviation = -6.387e-05; in-situ estimate (2.19) = -6.806e-05
C: (a), sigma=(-,-), highest b1={'M': 2.0, 'tau': 0.1, 'gamma': 0.025, 'sigma': -1, 'coupling': 'a'} b2={'M': 3.0, 'tau': 0.15, 'gamma': 0.02, 'sigma': -1, 'coupling': 'a'} centres=(3, 8) widths=(1.3, 1.3)
tau=0 gaps: delta_1=2.0039 delta_f=0.0917 delta_2=8.0124 delta=0.0917; body1 (delta_1=2.0064, delta_f=0.0958) body2 (delta_1=4.0075, delta_f=0.0953)
s=1.0: a=(3.785826e-07,-5.810415e-07) a_evol/a-1=(3.4e-10,-4.0e-09) v(0)=(-1.4e-15,3.1e-16)
a^w=(3.819916e-07,-5.730847e-07) eps_cut=a-a^w=(-3.41e-09,-7.96e-09) p_cut(0)=(2.0e-04,2.0e-04) max p_cut=(2.3e-04,2.7e-04)
a_pred=(3.779570e-07,-5.798805e-07) [free-packet term (-3.47e-09,-7.80e-09); exact a^(0)=(-3.48e-09,-7.83e-09)] E''=(-4.986555e-03,-7.478886e-03) dev=(1.655e-03,2.002e-03) dev_w=(1.675e-03,1.972e-03)
ratio exact=-0.651559 pred=-0.651784 -E''1/E''2=-0.666751; mutual ratio (a-a^(0))=-0.666525, /(-E''1/E''2)-1=-3.38e-04 (first-order pred: -3.17e-05) c=(0.7439,0.7439)
band gap=0.0879 band leak=1.2e-03 single (gap, leak)=(0.0954,5.9e-04),(0.0946,5.9e-04) tau'/delta_1=0.075 tau'^2/(delta_1 delta)=0.123
s=0.5: a=(9.517570e-08,-1.435804e-07) a_evol/a-1=(1.3e-09,-8.4e-11) v(0)=(4.8e-16,4.3e-15)
a^w=(9.537485e-08,-1.430610e-07) eps_cut=a-a^w=(-1.99e-10,-5.19e-10) p_cut(0)=(2.0e-04,2.0e-04) max p_cut=(2.1e-04,2.1e-04)
a_pred=(9.514152e-08,-1.435145e-07) [free-packet term (-2.17e-10,-4.87e-10); exact a^(0)=(-2.17e-10,-4.88e-10)] E''=(-1.246668e-03,-1.869808e-03) dev=(3.593e-04,4.594e-04) dev_w=(3.608e-04,4.557e-04)
ratio exact=-0.662874 pred=-0.662940 -E''1/E''2=-0.666736; mutual ratio (a-a^(0))=-0.666650, /(-E''1/E''2)-1=-1.29e-04 (first-order pred: -3.17e-05) c=(0.7439,0.7439)
band gap=0.0910 band leak=2.9e-04 single (gap, leak)=(0.0957,1.5e-04),(0.0951,1.5e-04) tau'/delta_1=0.037 tau'^2/(delta_1 delta)=0.031
s=0.25: a=(2.382741e-08,-3.579076e-08) a_evol/a-1=(8.8e-10,2.7e-09) v(0)=(-5.7e-15,-3.2e-15)
a^w=(2.383652e-08,-3.575247e-08) eps_cut=a-a^w=(-9.11e-12,-3.83e-11) p_cut(0)=(2.0e-04,2.0e-04) max p_cut=(2.0e-04,2.0e-04)
a_pred=(2.382615e-08,-3.578765e-08) [free-packet term (-1.35e-11,-3.05e-11); exact a^(0)=(-1.35e-11,-3.05e-11)] E''=(-3.116688e-04,-4.674576e-04) dev=(5.297e-05,8.691e-05) dev_w=(5.310e-05,8.620e-05)
ratio exact=-0.665742 pred=-0.665765 -E''1/E''2=-0.666732; mutual ratio (a-a^(0))=-0.666688, /(-E''1/E''2)-1=-6.54e-05 (first-order pred: -3.17e-05) c=(0.7439,0.7439)
band gap=0.0916 band leak=6.7e-05 single (gap, leak)=(0.0958,3.7e-05),(0.0952,3.7e-05) tau'/delta_1=0.019 tau'^2/(delta_1 delta)=0.008
extrapolated (s->0) dev_i = (-4.756e-05, -3.607e-05); in-situ estimate (2.16) = (-4.794e-05, -3.287e-05)
extrapolated ratio deviation = -1.149e-05; in-situ estimate (2.19) = -1.507e-05
D: (c), sigma=(+,+), lowest, unequal widths b1={'M': 2.0, 'tau': 0.1, 'gamma': 0.025, 'sigma': 1, 'coupling': 'c'} b2={'M': 3.0, 'tau': 0.15, 'gamma': 0.02, 'sigma': 1, 'coupling': 'c'} centres=(3, 8) widths=(1.1, 1.6)
tau=0 gaps: delta_1=2.0039 delta_f=0.0917 delta_2=8.0124 delta=0.0917; body1 (delta_1=2.0064, delta_f=0.0958) body2 (delta_1=4.0075, delta_f=0.0953)
s=1.0: a=(3.335546e-07,-6.974325e-07) a_evol/a-1=(1.5e-09,-3.0e-09) v(0)=(-7.5e-16,-7.0e-16)
a^w=(3.337905e-07,-6.219309e-07) eps_cut=a-a^w=(-2.36e-10,-7.55e-08) p_cut(0)=(1.2e-05,2.4e-03) max p_cut=(2.4e-05,2.5e-03)
a_pred=(3.330744e-07,-6.962943e-07) [free-packet term (-2.34e-10,-7.12e-08); exact a^(0)=(-2.37e-10,-7.13e-08)] E''=(5.002073e-03,7.497049e-03) dev=(1.442e-03,1.635e-03) dev_w=(1.449e-03,1.704e-03)
ratio exact=-0.478261 pred=-0.478353 -E''1/E''2=-0.667206; mutual ratio (a-a^(0))=-0.533063, /(-E''1/E''2)-1=-2.01e-01 (first-order pred: -2.01e-01) c=(0.6615,0.8228)
band gap=0.0879 band leak=1.2e-03 single (gap, leak)=(0.0954,5.9e-04),(0.0946,5.9e-04) tau'/delta_1=0.076 tau'^2/(delta_1 delta)=0.127
s=0.5: a=(8.334478e-08,-1.607986e-07) a_evol/a-1=(-3.3e-09,-1.7e-09) v(0)=(-1.9e-16,-1.9e-15)
a^w=(8.335921e-08,-1.552912e-07) eps_cut=a-a^w=(-1.44e-11,-5.51e-09) p_cut(0)=(1.2e-05,2.3e-03) max p_cut=(1.3e-05,2.4e-03)
a_pred=(8.331406e-08,-1.607293e-07) [free-packet term (-1.46e-11,-4.45e-09); exact a^(0)=(-1.47e-11,-4.45e-09)] E''=(1.250541e-03,1.874337e-03) dev=(3.687e-04,4.314e-04) dev_w=(3.691e-04,4.301e-04)
ratio exact=-0.518318 pred=-0.518350 -E''1/E''2=-0.667191; mutual ratio (a-a^(0))=-0.533169, /(-E''1/E''2)-1=-2.01e-01 (first-order pred: -2.01e-01) c=(0.6615,0.8228)
band gap=0.0910 band leak=2.9e-04 single (gap, leak)=(0.0957,1.5e-04),(0.0951,1.5e-04) tau'/delta_1=0.038 tau'^2/(delta_1 delta)=0.032
s=0.25: a=(2.083376e-08,-3.935311e-08) a_evol/a-1=(-1.1e-09,9.4e-09) v(0)=(2.7e-16,5.3e-17)
a^w=(2.083461e-08,-3.881124e-08) eps_cut=a-a^w=(-8.52e-13,-5.42e-10) p_cut(0)=(1.2e-05,2.3e-03) max p_cut=(1.2e-05,2.3e-03)
a_pred=(2.083136e-08,-3.934824e-08) [free-packet term (-9.15e-13,-2.78e-10); exact a^(0)=(-9.16e-13,-2.78e-10)] E''=(3.126368e-04,4.685891e-04) dev=(1.152e-04,1.240e-04) dev_w=(1.152e-04,1.222e-04)
ratio exact=-0.529406 pred=-0.529410 -E''1/E''2=-0.667188; mutual ratio (a-a^(0))=-0.533198, /(-E''1/E''2)-1=-2.01e-01 (first-order pred: -2.01e-01) c=(0.6615,0.8228)
band gap=0.0916 band leak=6.8e-05 single (gap, leak)=(0.0958,3.7e-05),(0.0952,3.7e-05) tau'/delta_1=0.019 tau'^2/(delta_1 delta)=0.008
extrapolated (s->0) dev_i = (3.203e-05, 2.095e-05); in-situ estimate (2.16) = (3.926e-05, 2.645e-05)
extrapolated ratio deviation = 1.107e-05; in-situ estimate (2.19) = 1.281e-05
E: (c), sigma=(+,+), lowest b1={'M': 2.0, 'tau': 0.1, 'gamma': 0.025, 'sigma': 1, 'coupling': 'c'} b2={'M': 3.0, 'tau': 0.15, 'gamma': 0.02, 'sigma': 1, 'coupling': 'c'} centres=(3, 8) widths=(1.3, 1.3)
tau=0 gaps: delta_1=2.0039 delta_f=0.0917 delta_2=8.0124 delta=0.0917; body1 (delta_1=2.0064, delta_f=0.0958) body2 (delta_1=4.0075, delta_f=0.0953)
s=1.0: a=(3.796931e-07,-5.823648e-07) a_evol/a-1=(1.2e-09,-3.7e-09) v(0)=(-8.4e-16,-8.7e-16)
a^w=(3.831225e-07,-5.743722e-07) eps_cut=a-a^w=(-3.43e-09,-7.99e-09) p_cut(0)=(2.0e-04,2.0e-04) max p_cut=(2.3e-04,2.7e-04)
a_pred=(3.791223e-07,-5.813077e-07) [free-packet term (-3.49e-09,-7.84e-09); exact a^(0)=(-3.50e-09,-7.87e-09)] E''=(5.002073e-03,7.497049e-03) dev=(1.505e-03,1.818e-03) dev_w=(1.524e-03,1.790e-03)
ratio exact=-0.651985 pred=-0.652189 -E''1/E''2=-0.667206; mutual ratio (a-a^(0))=-0.667006, /(-E''1/E''2)-1=-2.99e-04 (first-order pred: -3.17e-05) c=(0.7439,0.7439)
band gap=0.0879 band leak=1.2e-03 single (gap, leak)=(0.0954,5.9e-04),(0.0946,5.9e-04) tau'/delta_1=0.076 tau'^2/(delta_1 delta)=0.127
s=0.5: a=(9.547296e-08,-1.439284e-07) a_evol/a-1=(1.1e-09,-1.6e-09) v(0)=(-3.7e-16,-1.5e-15)
a^w=(9.567333e-08,-1.434067e-07) eps_cut=a-a^w=(-2.00e-10,-5.22e-10) p_cut(0)=(2.0e-04,2.0e-04) max p_cut=(2.1e-04,2.1e-04)
a_pred=(9.543647e-08,-1.438633e-07) [free-packet term (-2.18e-10,-4.90e-10); exact a^(0)=(-2.18e-10,-4.90e-10)] E''=(1.250541e-03,1.874337e-03) dev=(3.823e-04,4.522e-04) dev_w=(3.831e-04,4.498e-04)
ratio exact=-0.663337 pred=-0.663383 -E''1/E''2=-0.667191; mutual ratio (a-a^(0))=-0.667125, /(-E''1/E''2)-1=-9.88e-05 (first-order pred: -3.17e-05) c=(0.7439,0.7439)
band gap=0.0910 band leak=2.9e-04 single (gap, leak)=(0.0957,1.5e-04),(0.0951,1.5e-04) tau'/delta_1=0.038 tau'^2/(delta_1 delta)=0.032
s=0.25: a=(2.390296e-08,-3.587884e-08) a_evol/a-1=(1.3e-10,1.6e-10) v(0)=(7.0e-16,3.2e-16)
a^w=(2.391213e-08,-3.584039e-08) eps_cut=a-a^w=(-9.17e-12,-3.84e-11) p_cut(0)=(2.0e-04,2.0e-04) max p_cut=(2.0e-04,2.0e-04)
a_pred=(2.390011e-08,-3.587436e-08) [free-packet term (-1.36e-11,-3.06e-11); exact a^(0)=(-1.36e-11,-3.06e-11)] E''=(3.126368e-04,4.685891e-04) dev=(1.190e-04,1.249e-04) dev_w=(1.189e-04,1.246e-04)
ratio exact=-0.666213 pred=-0.666217 -E''1/E''2=-0.667188; mutual ratio (a-a^(0))=-0.667163, /(-E''1/E''2)-1=-3.74e-05 (first-order pred: -3.17e-05) c=(0.7439,0.7439)
band gap=0.0916 band leak=6.8e-05 single (gap, leak)=(0.0958,3.7e-05),(0.0952,3.7e-05) tau'/delta_1=0.019 tau'^2/(delta_1 delta)=0.008
extrapolated (s->0) dev_i = (3.278e-05, 1.709e-05); in-situ estimate (2.16) = (4.794e-05, 3.287e-05)
extrapolated ratio deviation = 1.569e-05; in-situ estimate (2.19) = 1.507e-05
A: with gamma_1, gamma_2 times 0.5 b1={'M': 2.0, 'tau': 0.1, 'gamma': 0.0125, 'sigma': 1, 'coupling': 'a'} b2={'M': 3.0, 'tau': 0.15, 'gamma': 0.01, 'sigma': 1, 'coupling': 'a'} centres=(3, 8) widths=(1.3, 1.3)
tau=0 gaps: delta_1=2.0027 delta_f=0.0952 delta_2=8.0069 delta=0.0952; body1 (delta_1=2.0034, delta_f=0.0973) body2 (delta_1=4.0040, delta_f=0.0970)
s=1.0: a=(9.947245e-08,-1.623294e-07) a_evol/a-1=(6.6e-10,-8.1e-09) v(0)=(1.4e-17,7.7e-16)
a^w=(1.029430e-07,-1.544380e-07) eps_cut=a-a^w=(-3.47e-09,-7.89e-09) p_cut(0)=(2.0e-04,2.0e-04) max p_cut=(2.3e-04,2.7e-04)
a_pred=(9.940714e-08,-1.621638e-07) [free-packet term (-3.48e-09,-7.83e-09); exact a^(0)=(-3.49e-09,-7.85e-09)] E''=(4.994293e-03,7.491741e-03) dev=(6.571e-04,1.022e-03) dev_w=(7.631e-04,8.757e-04)
ratio exact=-0.612781 pred=-0.613005 -E''1/E''2=-0.666640; mutual ratio (a-a^(0))=-0.666540, /(-E''1/E''2)-1=-1.50e-04 (first-order pred: -3.00e-05) c=(0.7439,0.7439)
band gap=0.0909 band leak=1.2e-03 single (gap, leak)=(0.0969,5.9e-04),(0.0963,5.9e-04) tau'/delta_1=0.075 tau'^2/(delta_1 delta)=0.118
s=0.5: a=(2.550794e-08,-3.908250e-08) a_evol/a-1=(1.0e-08,1.4e-09) v(0)=(8.2e-16,1.0e-15)
a^w=(2.572040e-08,-3.858404e-08) eps_cut=a-a^w=(-2.12e-10,-4.98e-10) p_cut(0)=(2.0e-04,2.0e-04) max p_cut=(2.0e-04,2.1e-04)
a_pred=(2.550402e-08,-3.907413e-08) [free-packet term (-2.17e-10,-4.89e-10); exact a^(0)=(-2.18e-10,-4.90e-10)] E''=(1.248581e-03,1.872958e-03) dev=(1.539e-04,2.142e-04) dev_w=(1.607e-04,2.032e-04)
ratio exact=-0.652669 pred=-0.652708 -E''1/E''2=-0.666636; mutual ratio (a-a^(0))=-0.666586, /(-E''1/E''2)-1=-7.43e-05 (first-order pred: -3.00e-05) c=(0.7439,0.7439)
band gap=0.0944 band leak=3.0e-04 single (gap, leak)=(0.0972,1.5e-04),(0.0969,1.5e-04) tau'/delta_1=0.037 tau'^2/(delta_1 delta)=0.029
s=0.25: a=(6.416962e-09,-9.677293e-09) a_evol/a-1=(-9.1e-09,-3.1e-08) v(0)=(7.9e-16,-5.1e-15)
a^w=(6.429273e-09,-9.644506e-09) eps_cut=a-a^w=(-1.23e-11,-3.28e-11) p_cut(0)=(2.0e-04,2.0e-04) max p_cut=(2.0e-04,2.0e-04)
a_pred=(6.416770e-09,-9.676855e-09) [free-packet term (-1.36e-11,-3.06e-11); exact a^(0)=(-1.36e-11,-3.06e-11)] E''=(3.121457e-04,4.682410e-04) dev=(2.988e-05,4.524e-05) dev_w=(3.034e-05,4.418e-05)
ratio exact=-0.663095 pred=-0.663105 -E''1/E''2=-0.666635; mutual ratio (a-a^(0))=-0.666605, /(-E''1/E''2)-1=-4.43e-05 (first-order pred: -3.00e-05) c=(0.7439,0.7439)
band gap=0.0951 band leak=7.2e-05 single (gap, leak)=(0.0973,3.7e-05),(0.0970,3.7e-05) tau'/delta_1=0.019 tau'^2/(delta_1 delta)=0.007
extrapolated (s->0) dev_i = (-1.131e-05, -8.144e-06); in-situ estimate (2.16) = (-1.158e-05, -7.835e-06)
extrapolated ratio deviation = -3.165e-06; in-situ estimate (2.19) = -3.741e-06
E: with gamma_1, gamma_2 times 0.5 b1={'M': 2.0, 'tau': 0.1, 'gamma': 0.0125, 'sigma': 1, 'coupling': 'c'} b2={'M': 3.0, 'tau': 0.15, 'gamma': 0.01, 'sigma': 1, 'coupling': 'c'} centres=(3, 8) widths=(1.3, 1.3)
tau=0 gaps: delta_1=2.0027 delta_f=0.0952 delta_2=8.0069 delta=0.0952; body1 (delta_1=2.0034, delta_f=0.0973) body2 (delta_1=4.0040, delta_f=0.0970)
s=1.0: a=(9.961721e-08,-1.625282e-07) a_evol/a-1=(4.0e-10,-5.1e-09) v(0)=(1.5e-15,7.7e-16)
a^w=(1.030990e-07,-1.546167e-07) eps_cut=a-a^w=(-3.48e-09,-7.91e-09) p_cut(0)=(2.0e-04,2.0e-04) max p_cut=(2.3e-04,2.7e-04)
a_pred=(9.956685e-08,-1.623873e-07) [free-packet term (-3.49e-09,-7.85e-09); exact a^(0)=(-3.50e-09,-7.88e-09)] E''=(5.002608e-03,7.501592e-03) dev=(5.058e-04,8.675e-04) dev_w=(6.143e-04,7.181e-04)
ratio exact=-0.612923 pred=-0.613144 -E''1/E''2=-0.666873; mutual ratio (a-a^(0))=-0.666784, /(-E''1/E''2)-1=-1.33e-04 (first-order pred: -3.00e-05) c=(0.7439,0.7439)
band gap=0.0909 band leak=1.2e-03 single (gap, leak)=(0.0969,5.9e-04),(0.0963,5.9e-04) tau'/delta_1=0.076 tau'^2/(delta_1 delta)=0.120
s=0.5: a=(2.554948e-08,-3.913329e-08) a_evol/a-1=(3.4e-09,1.0e-09) v(0)=(7.1e-16,-9.1e-17)
a^w=(2.576264e-08,-3.863358e-08) eps_cut=a-a^w=(-2.13e-10,-5.00e-10) p_cut(0)=(2.0e-04,2.0e-04) max p_cut=(2.0e-04,2.1e-04)
a_pred=(2.554609e-08,-3.912609e-08) [free-packet term (-2.18e-10,-4.90e-10); exact a^(0)=(-2.18e-10,-4.91e-10)] E''=(1.250658e-03,1.875418e-03) dev=(1.330e-04,1.842e-04) dev_w=(1.392e-04,1.741e-04)
ratio exact=-0.652884 pred=-0.652917 -E''1/E''2=-0.666869; mutual ratio (a-a^(0))=-0.666826, /(-E''1/E''2)-1=-6.46e-05 (first-order pred: -3.00e-05) c=(0.7439,0.7439)
band gap=0.0944 band leak=3.0e-04 single (gap, leak)=(0.0972,1.5e-04),(0.0969,1.5e-04) tau'/delta_1=0.038 tau'^2/(delta_1 delta)=0.030
s=0.25: a=(6.427681e-09,-9.690065e-09) a_evol/a-1=(1.7e-08,1.6e-08) v(0)=(-2.0e-16,4.2e-16)
a^w=(6.440034e-09,-9.657198e-09) eps_cut=a-a^w=(-1.24e-11,-3.29e-11) p_cut(0)=(2.0e-04,2.0e-04) max p_cut=(2.0e-04,2.0e-04)
a_pred=(6.427421e-09,-9.689599e-09) [free-packet term (-1.36e-11,-3.06e-11); exact a^(0)=(-1.36e-11,-3.07e-11)] E''=(3.126649e-04,4.688557e-04) dev=(4.049e-05,4.809e-05) dev_w=(4.073e-05,4.738e-05)
ratio exact=-0.663327 pred=-0.663332 -E''1/E''2=-0.666868; mutual ratio (a-a^(0))=-0.666844, /(-E''1/E''2)-1=-3.65e-05 (first-order pred: -3.00e-05) c=(0.7439,0.7439)
band gap=0.0951 band leak=7.2e-05 single (gap, leak)=(0.0973,3.7e-05),(0.0970,3.7e-05) tau'/delta_1=0.019 tau'^2/(delta_1 delta)=0.008
extrapolated (s->0) dev_i = (9.724e-06, 5.820e-06); in-situ estimate (2.16) = (1.158e-05, 7.835e-06)
extrapolated ratio deviation = 3.907e-06; in-situ estimate (2.19) = 3.741e-06