Annila and Stanley suggest that the thermodynamic equivalent velocities, ‘v_j’, (rate of change of commodities) depend on their ‘conductance’ (‘sigma_jk’) and ‘free energy’ (‘A_jk’) (eq.5). The idea is that conductance is an abstract of production capacity and free energy is the driving force of economic activities (will become clearer with further investigation).
Where ‘k_B T’ is the Boltzmann constant times the average temperature of the system, assuming ‘energy’ is conserved, this is taken to be a constant.
A visualisation of ‘v_j’, ‘sigma_jk’, and ‘A_jk’ (eq.6) should help understand how the components come together. Do not take this as a lesson on indexes (the subscripts), as this description is pertinent to nested lists in python, the matrices seen may be transposed (flipped along the diagonal, e.g., ‘A_j(k-1)’ swaps with ‘A_(j-1)k’), but their positions will align with how they appear typed in the code.
The subscripts denote the property in the transfer from ‘N_j’ (a material) to ‘N_k’ (a product). If, for example, we wanted to consider the conversion between ‘N_3’ (written in python as N[3], matching because we started our matrices at 0,0) and ‘N_5’, we could be interested in ‘v_3’ and ‘v_5’, but we may not find that they are equal and opposite due to other contributions from alternate commodities. Instead, we can find ‘v_35’ using the equation for ‘v_jk’ (eq.5), (eq.7).
Where, if ‘v_35<0’, ‘N_3’ is increasing at the expense of ‘N_5’ and vice versa. We found it easier to read, e.g., ‘v_35’, as “velocity 3 to 5” which implies a directionality towards a positive value.
Combining ‘sigma_jk’ and ‘A_jk’ to create a ‘v_jk’ matrix and then a ‘v_j’ (eq.5) column vector (eq.8) incorporates all ideas we have explored thus far.
The conductance of from one commodity to another is thermoeconomically considered to be the production capacity so this is as ‘pure’ as our code can be with respect to Annila & Salthe. To utilise the model, one would have to go out and determine these figures physically. Our interpretation of free energy thus far, however, can be broken down farther into its constituent parts (eq.9).
Where ‘mu_k’, ‘mu_j’ are the potentials of their respective commodities, the article is somewhat unclear as to the thermoeconomics equivalent, but the potential appears to be the ‘cost’ of the commodity taking this form (like storing nuclear materials would be more expensive than wood). ‘Delta Q_jk’ is the effort/cost required to convert from ‘N_j’ to ‘N_k’.
The potentials ‘mu_j’, ‘mu_k’ are themselves defined in terms of energy density ‘phi_j’, ‘phi_k’ . These energy densities are determined from a Gibbs energy which is stated to be the energy content (which we are interpreting as worth) relative to ‘k_B T’ (eq.10).
If one is unfamiliar with Gibbs then it is likely more intuitive to focus on the commodity’s worth as, unless an attempt at a real-life model is being made, it is fairly useless in figuring out how the pieces fit together.
The Evolution of Physics in Finance
Completed by: Jake Wilkes, James Penston, Sam Jones and James Lundie.
© 2022 Physics in society project
