## Smooth transitions between two constants

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Suppose we have a parameter that has two different values depending on the value of a dimensionless number. For example when the dimensionless number is much less than 1, x = 2/3, and when x is much greater than 1, x = 1. We desire a smooth transition from 2/3 to 1 as a function of x to avoid discontinuities in functions of x. We will adapt the smooth transitions between functions to be a smooth transition between constants.

We define our function as $$x(D) = x0 + (x1 - x0)*(1 - sigma(D,w)$$. We control the rate of the transition by the variable $$w$$

import numpy as np
import matplotlib.pyplot as plt

x0 = 2.0 / 3.0
x1 = 1.5

w = 0.05

D = np.linspace(0,2, 500)

sigmaD = 1.0 / (1.0 + np.exp(-(1 - D) / w))

x =  x0 + (x1 - x0)*(1 - sigmaD)

plt.plot(D, x)
plt.xlabel('D'); plt.ylabel('x')
plt.savefig('images/smooth-transitions-constants.png')


This is a nice trick to get an analytical function with continuous derivatives for a transition between two constants. You could have the transition occur at a value other than D = 1, as well by changing the argument to the exponential function.