Deriving PLA from MaxCal

The principle of least action (PLA), or more accurately, the principle of stationary action, is one first principle in physics. PLA offers the deepest explanatory power of our external reality. In this blog, I shall summarise a beautiful paper “Principle of maximum caliber and quantum physics” [1]. I shall omit the part of quantum physics but show how to derive PLA from the principle of Maximum Calibre (MaxCal), the generalisation of MaxEnt.

Let a and b be the two points in the phase space of a dynamical system, and p_i(ab) be the probability of the system will go from a to b following an individual path i, then the calibre is

\begin{aligned}  S(a,b) = -\sum_{i=1}^{N} p_i(ab) \ln p_i(ab) \end{aligned} \ \ \ \ (1),

where N is the total number of possible paths from point a to point b. We assume A_i(a,b) is the physical property that characterises the individual path i and the average of all possible path is \langle A(a,b) \rangle. We also assume the sum of probabilities of all path is 1. The two assumptions can be written as:

\begin{aligned} \sum_i^{N} p_i(ab) A_i(a,b) = \langle A(a,b) \rangle \end{aligned} \ \ \ \ (2)

\begin{aligned} \sum_i^N p_i(ab) = 1\end{aligned} \ \ \ \ (3)

Using the Lagrange multipliers method, the auxiliary function can be written as

\begin{aligned} S^{'} = -\sum_i^N p_i(ab) \ln p_i(ab) - \eta \left( \sum_i^N p_i(ab) A_i(a,b) - \langle A(a,b) \rangle \right) -   \lambda \left( \sum_i^N p_i(ab) - 1 \right) \end{aligned} \ \ \ \ (4).

To obtain the stationary points, we calculate the derivatives of S' with respect to p_i and the two multipliers \lambda and \eta:

\begin{aligned} \frac{\partial S^{'}}{\partial p_i} = -\ln p_i(ab) -1 - \eta  A_i(a,b) - \lambda = 0 \end{aligned} \ \ \ \ (5)

\begin{aligned} \frac{\partial S^{'}}{\partial \eta} = \sum_i^N p_i(ab) A_i(a,b) - \langle A(a,b) \rangle = 0 \end{aligned} \ \ \ \ (6)

\begin{aligned} \frac{\partial S^{'}}{\partial \lambda} = \sum_i^N p_i(ab) - 1 = 0 \end{aligned} \ \ \ \ (7)

From equation (5), we have

\begin{aligned} p_i(ab) =  \exp(- 1 - \lambda - \eta A_i(a,b))   = \exp(-1-\lambda) \exp (-\eta A_i(a,b))  \end{aligned} \ \ \ \ (8)

From equation (7), we have

\begin{aligned} \sum_i^N p_i(ab) = \sum_i^N \exp (-1- \eta A_i(a,b) - \lambda) = \exp (-1-\lambda)  \sum_i^N \exp (-\eta A_i(a,b)) = 1 \end{aligned}, which yields:

\begin{aligned}  \exp(-1-\lambda) = \frac{1}{ \sum_i^N \exp (-\eta A_i(a,b)) } \end{aligned} \ \ \ \ (9)

We then substitute \exp(-1-\lambda) in equation (8) with equation (9) to obtain

\begin{aligned} p_i(ab) = \frac{1}{Z} \exp(-\eta A_i(a,b)) \end{aligned} \ \ \ \ (10) ,

where Z = \sum_{i=1}^{N} \exp(-\eta A_i(a,b))

From equation (10), we notice that p_i(ab) , i.e., the probability of the ith path between points a and b , is maximum when its corresponding physical property A_i(a,b) is minimum.

If we replace A_i(a,b) with the word “action”, then we derive the principle of least action. However, we should stress that MaxCal is more general since it does not only contains the least action principle as its most probable outcome but also allows other trajectories with lower probability. Those trajectories with lower probability are not of use But, since classically there is only one allowed trajectory, we can infer that η must be high, thus suppressing the extra trajectories, and leaving only the most probable one.

1. General IJ. Principle of maximum caliber and quantum physics. Phys Rev E. 2018;98: 012110.

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