# Hadron–Quark Combustion as a Nonlinear, Dynamical System

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## Abstract

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## 1. Introduction

## 2. Feedback Effects and the Reaction Zone

**Flavor equilibration**: The conversion of two flavoured quark matter to three flavoured quark matter through the reactions (1)–(3) releases binding energy in the form of heat, increasing the temperature behind the front. The increase of temperature stiffens the quark EoS, increasing the pressure behind the front and therefore accelerating the burning speed.**Neutrino pressure**: Neutrinos deposit momentum into the reaction zone, accelerating the interface into faster speeds.**Loss of lepton number**: Neutrinos, as they diffuse from higher to lower chemical potentials, deposit the chemical potential difference in the form of heat. This heat increases the temperature and therefore enhances the pressure behind the interface. This phenomenon is very similar to what is referred as Joule heating in papers concerning proto–neutron star evolution (e.g., [18]).

**Electron pressure**: Electron capture “eats up” the electrons behind the interface, generating a large electron gradient (see the electron Fermi momentum distribution in Figure 1a). These electron gradients generate a degeneracy pressure that pushes the interface backwards, decelerating the burning front. See Figure 1b for a graphical representation of the electron pressure gradient.**Neutrino cooling**: Neutrinos that escape from the burning front carry energy away from the reaction zone, which reduces the temperature and therefore the pressure behind the interface. This quenching effect was first detailed in Paper I.

## 3. Leptons and Positive Feedback

## 4. Conclusions

- Assuming the system is steady-state, in other words, equating all temporal derivatives to zero.
- Assuming that the front is in pressure equilibrium, that is, fixing $\nabla P=0$.
- The above two points lead to the cancellation of the important nonlinearities. Pressure equilibrium and a steady-state momentum make the fluid velocity a constant in space and time.
- Another related pitfall is collapsing the rich structure of the reaction zone into a discontinuity by solving the jump conditions instead of the continuous hydrodynamic equations. This also leads to a steady-state solution, which eliminates the dynamism of the system.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**(

**a**): Simulation snapshot of the burning interface. ${p}_{F,i}$ are the Fermi momenta for particles i, and T is the temperature. In both panels, the interface lies at position zero depicted by the vertical line. The arrows represent the directions of the force vectors and their labels depict the processes that caused them. Upstream is the side behind (left side of the vertical line) the interface, and downstream is the side in front of it (right side of the vertical line). (

**b**): The pressure gradients for the leptons and quarks shown in Figure 1a. Figure and caption were taken from Paper II [17].

**Figure 2.**Distance travelled by the combustion front as a function of time from a numerical simulation. The line labeled as “free streaming” represents the burning front with neutrinos free streaming, while the line labeled as “trapped” plots the burning front with trapped neutrinos. Notice how the front halts for the remainder of the simulation for the free streaming case. The thermodynamic parameters for the simulation were an initial temperature of T = 20 MeV, an initial lepton fraction of Y${}_{L}=0.2$, and an initial baryonic density of ${n}_{B}=0.35$ fm${}^{-3}$.

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**MDPI and ACS Style**

Ouyed, A.; Ouyed, R.; Jaikumar, P.
Hadron–Quark Combustion as a Nonlinear, Dynamical System. *Universe* **2018**, *4*, 51.
https://doi.org/10.3390/universe4030051

**AMA Style**

Ouyed A, Ouyed R, Jaikumar P.
Hadron–Quark Combustion as a Nonlinear, Dynamical System. *Universe*. 2018; 4(3):51.
https://doi.org/10.3390/universe4030051

**Chicago/Turabian Style**

Ouyed, Amir, Rachid Ouyed, and Prashanth Jaikumar.
2018. "Hadron–Quark Combustion as a Nonlinear, Dynamical System" *Universe* 4, no. 3: 51.
https://doi.org/10.3390/universe4030051