Sharma Mittal Entropy Gravity

---
name: sharma-mittal-entropy-gravity
description: "Sharma-Mittal entropy framework bridging information theory, black hole thermodynamics, and infrared gravity modifications. Derives modified gravitational force laws from generalized entropy, reproduces MOND-like regime. Activates: sharma-mittal entropy, generalized entropy, emergent gravity, MOND, black hole thermodynamics, information bounds, infrared gravity, entropic gravity"
metadata:
  arxiv_id: "2606.15996"
  published: "2026-06-14"
  authors: "Abdelhakim Benkrane, Giuseppe Gaetano Luciano, Ahmad Sheykhi"
  tags: [information-theory, gravity, thermodynamics, black-hole, MOND, emergent-gravity]
---

# Sharma-Mittal Entropy Framework for Gravity

## Description

Sharma-Mittal (SM) entropy provides a two-parameter generalization encompassing both Renyi and Tsallis frameworks. This methodology connects generalized entropy to gravitational physics through black hole thermodynamics and entropic gravity.

## Activation Keywords
- sharma-mittal entropy
- generalized entropy gravity
- emergent gravity MOND
- black hole thermodynamics
- entropic gravity modified
- information bounds gravity
- infrared gravity modification

## Core Methodology

### Framework Components

1. **Sharma-Mittal Entropy**: Two-parameter (R, delta) generalization that interpolates between Renyi and Tsallis entropies in appropriate limits

2. **Bekenstein Bound Compatibility**: Gravitational realization of SM entropy consistently interpolates between Renyi and Bekenstein-Hawking entropies

3. **Landauer's Principle Extension**: Modified mass-loss relation from one-bit information erasure with parameter-dependent asymptotic behavior in small- and large-mass regimes

4. **Entropic Gravity Modification**: Within Verlinde's entropic gravity framework, derive modified gravitational force and acceleration laws induced by SM entropy

### Key Results

- Modified acceleration deviates from Newtonian prediction at large distances
- Naturally reproduces MOND-like regime for parameter relation R/delta = 3/2
- Direct connection between SM entropy parameters and MOND acceleration scale a_0
- Links black hole thermodynamics, information theory, and infrared gravity modifications

## Usage Patterns

### Pattern 1: Entropic Gravity Derivation
When studying how generalized entropy frameworks modify gravitational laws:
1. Define the generalized entropy function S_SM(m, R, delta)
2. Apply Landauer's principle: delta M = T delta S
3. Derive modified mass-loss/acceleration relations
4. Compare with Newtonian prediction at various distance scales
5. Identify parameter regimes reproducing observed phenomena (e.g., MOND)

### Pattern 2: Black Hole Thermodynamics Analysis
When analyzing black hole entropy with generalized statistical frameworks:
1. Check compatibility with Bekenstein bound
2. Compute entropy in appropriate limits (R -> 0, delta -> 0)
3. Verify interpolation between known entropy forms
4. Study asymptotic behavior in small/large mass regimes

### Pattern 3: Information-Gravity Connection
When exploring how information-theoretic quantities manifest in gravitational physics:
1. Identify the information-theoretic measure (entropy, mutual information)
2. Map to thermodynamic quantities via Landauer/Verlinde frameworks
3. Derive gravitational consequences
4. Test against observational constraints

## Related Skills
- `quantum-information-science` - broader information theory context
- `quantum-fisher-information-duality` - QFI as information-theoretic measure
- `quantum-metabolic-neuroimaging-limit` - fundamental information bounds