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A new family of ladder operators for macroscopic systems, with applications

Fabio Bagarello

Abstract

In a series of recent scientific contributions the role of bosonic and fermionic ladder operators in a macroscopic realm has been investigated. Creation, annihilation and number operators have been used in very different contexts, all sharing the same common main feature, i.e. the relevance of {\em discrete changes} in the description of the system. The main problem when using this approach is that computations are easy for Hamiltonians which are quadratic in the ladder operators, but become very complicated, both at the analytical and at the numerical level, when the Hamiltonian is not quadratic. In this paper we propose a possible alternative approach, again based on some sort of ladder operators, but for which an analytic solution can often be deduced without particular difficulties. We describe our proposal with few applications, mostly related to different versions of a predator-prey model, and to love affairs (from a decision-making point of view).

A new family of ladder operators for macroscopic systems, with applications

Abstract

In a series of recent scientific contributions the role of bosonic and fermionic ladder operators in a macroscopic realm has been investigated. Creation, annihilation and number operators have been used in very different contexts, all sharing the same common main feature, i.e. the relevance of {\em discrete changes} in the description of the system. The main problem when using this approach is that computations are easy for Hamiltonians which are quadratic in the ladder operators, but become very complicated, both at the analytical and at the numerical level, when the Hamiltonian is not quadratic. In this paper we propose a possible alternative approach, again based on some sort of ladder operators, but for which an analytic solution can often be deduced without particular difficulties. We describe our proposal with few applications, mostly related to different versions of a predator-prey model, and to love affairs (from a decision-making point of view).

Paper Structure

This paper contains 12 sections, 60 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Standard ladder operators
  3. Fermionic operators
  4. Bosonic operators
  5. New ladder operators
  6. A quadratic predator-prey model
  7. What now?
  8. A cubic predator-prey model
  9. Another cubic predator-prey model
  10. An interlude: an approximated solution
  11. Looking for an exact solution
  12. A different view to our systems and conclusions

Figures (1)

  • Figure 1: $x_1(t)$ (continuous line) and $x_3(t)$ (dot-dashed line) for the two agents model (up) and $x_1(t)$ (continuous line), $x_2(t)$ (dashed line) and $x_3(t)$ (dot-dashed line) for the three agents model (down). Constants are as follows: $\lambda=3$, $\Omega=2$ (up); $\lambda_1=2$, $\lambda_2$, $\Omega_1=2$, $\Omega_2=3$ (down, left); $\lambda_1=2$, $\lambda_2$, $\Omega_1=1$, $\Omega_2=4$ (down, right).