r"""
Flory's mean-field exponent vs. the renormalization-group value
=======================================================================

:meth:`~chemistrykit.polymer.systems.chain_statistics.RealChain.good_solvent`
uses Flory's original 1953 mean-field good-solvent exponent
(:math:`\nu=3/5`);
:meth:`~chemistrykit.polymer.systems.chain_statistics.RealChain.good_solvent_renormalization_group`
uses de Gennes' 1979 renormalization-group-refined value
(:math:`\nu\approx0.588`). The two *exponents* differ by only about 2%,
but because :math:`R\sim bn^\nu` is a power law, that small an exponent
difference compounds with chain length: the two chain-size curves are
nearly indistinguishable on a log-log plot for short chains, yet the
relative difference between the predicted `R` values grows steadily as
`n` increases -- exactly why precise critical exponents matter most for
long-chain, asymptotic behavior.
"""

# %%
import matplotlib.pyplot as plt
import numpy as np

from chemistrykit.polymer.systems.chain_statistics import RealChain
from chemistrykit.polymer.visualizers.polymer_plots import plot_chain_scaling

b = 0.5  # segment length
n_values = np.logspace(1, 5, 30)

flory = RealChain.good_solvent()
de_gennes = RealChain.good_solvent_renormalization_group()

fig, axes = plt.subplots(1, 2, figsize=(11, 4.5))

plot_chain_scaling(
    {"Flory mean-field (nu=3/5)": flory, "de Gennes RG (nu~=0.588)": de_gennes},
    n_values,
    b,
    ax=axes[0],
)

R_flory = flory.end_to_end_distance(n_values, b)
R_de_gennes = de_gennes.end_to_end_distance(n_values, b)
percent_difference = 100.0 * (R_de_gennes - R_flory) / R_flory

axes[1].semilogx(n_values, percent_difference, color="steelblue")
axes[1].set_xlabel("n (segments)")
axes[1].set_ylabel("(R_de_gennes - R_flory) / R_flory  (%)")
axes[1].set_title("Relative difference vs. chain length")
fig.tight_layout()

plt.show()

# %%
# Flory's simple mean-field exponent is within about 2% of the exact
# renormalization-group value -- remarkably accurate for an estimate
# that predates the renormalization-group machinery used to check it by
# over two decades. But the right panel shows that 2% exponent gap is
# not the whole story: since chain size is a power of `n`, the two
# models' predicted sizes diverge further apart the longer the chain,
# from a few percent at n~10 to well over 10% by n~10^4 -- a small
# difference in a critical exponent has an outsized effect precisely in
# the long-chain limit where it is meant to apply.