# Linear and Quadratic GUP, Liouville Theorem, Cosmological Constant, and Brick Wall Entropy

## TL;DR

This paper explores the combined effects of linear and quadratic Generalized Uncertainty Principles on equations of motion, phase space volume, and entropy, with implications for cosmology and black hole physics.

## Contribution

It extends previous GUP studies by analyzing the combined linear and quadratic terms and their impact on phase space, Liouville theorem, and entropy calculations.

## Key findings

- Modified equations of motion under combined GUP
- Derived an analogue of Liouville theorem with GUP
- Discussed implications for cosmological constant and black hole entropy

## Abstract

Motivated by the works on Equivalence Principle in the context of linear Generalized Uncertainty Principle and, independently, in the context of quadratic Generalized Uncertainty Principle, we expand these endeavors in the context of Generalized Uncertainty Principle when both linear and quadratic terms in momentum are include. We demonstrate how the definitions of equations of motion change upon that expansion. We also show how to obtain an analogue of Liouville theorem in the presence of linear and quadratic Generalized Uncertainty Principle. We employ the corresponding modified invariant unit volume of phase space to discuss the resulting density of states, the problem of cosmological constant, the black body radiation in curved spacetime, the concurrent energy and consequent no Brick Wall entropy.