# Non-Thermal Fixed Points in Bose Gas Experiments

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## Abstract

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## 1. Introduction

## 2. Universal Scaling and NTFPs

#### Global Observables

## 3. Experiments

#### 3.1. One-Dimensional Bose gas

#### 3.2. Spinor Bose Gas

#### 3.3. Homogeneous Three-Dimensional Bose Gas

#### 3.4. Harmonically Trapped Three-Dimensional Bose gas

## 4. Final Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

NTFP | non-thermal fixed point |

BEC | Bose–Einstein condensate |

IR | infrared |

UV | ultraviolet |

## References

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**Figure 1.**Illustration of the dynamics of a system passing near an NTFP. For several initial conditions (the key idea being that no fine-tuning is needed), the system can pass near a non-thermal fixed point. The correlation functions show a spatio-temporal scaling with a universal function when that occurs. After some time, the system leaves the vicinity of the NTFP and reaches equilibrium.

**Figure 2.**Universal scaling dynamics observed by the authors of [18]. (

**a**) Time evolution of the normalized momentum distributions. (

**b**) Momentum distributions scaled according to Equation (2). All the curves collapse into a single function, signaling the universal scaling. Reprinted by permission from Springer Nature Customer Service Centre GmbH: Springer Nature, Nature, Erne et al., Universal dynamics in an isolated one-dimensional Bose gas far from equilibrium, © 2018.

**Figure 3.**Universal scaling in a spinor Bose gas [19]. (

**a**) Time evolution of the structure factor (Equation (5)). (

**b**) After the scaling of Equation (2) has been applied, the curves collapse into a universal function. Reprinted by permission from Springer Nature Customer Service Centre GmbH: Springer Nature, Nature, Prüfer et al., Observation of universal dynamics in a spinor Bose gas far from equilibrium, © 2018.

**Figure 4.**Universal bidirectional scaling in a homogeneous Bose gas [20]. (

**a**) Momentum distributions as a function of time. (

**b**) Scaling provided by Equation (2) with $\alpha =-0.70\left(7\right)$ and $\beta =-0.14\left(2\right)$, which collapses the curves into a universal function for the UV region. (

**c**) The top panel shows the low-momenta region of the momentum distributions, while the bottom one depicts the scaling of Equation (2) with $\alpha =1.15\left(8\right)$ and $\beta =0.34\left(5\right)$. Reprinted by permission from Springer Nature Customer Service Centre GmbH: Springer Nature, Nature Physics, Glidden et al., Bidirectional dynamic scaling in an isolated Bose gas far from equilibrium, © 2021.

**Figure 6.**(

**a**) Three-dimensional momentum distribution reconstructed with the inverse Abel transform. (

**b**) Scaling provided by Equation (2) with $\alpha =-0.50$ and $\beta =-0.2$, which are the same exponents as those employed in the two-dimensional projection. (

**c**) Scaling using $\alpha =-0.75$ and $\beta =-0.2$, corresponding to the prediction of the exponents for the three-dimensional case. The collapse is much better than that shown in panel (

**b**). The figure is taken from [21].

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Madeira, L.; Bagnato, V.S.
Non-Thermal Fixed Points in Bose Gas Experiments. *Symmetry* **2022**, *14*, 678.
https://doi.org/10.3390/sym14040678

**AMA Style**

Madeira L, Bagnato VS.
Non-Thermal Fixed Points in Bose Gas Experiments. *Symmetry*. 2022; 14(4):678.
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**Chicago/Turabian Style**

Madeira, Lucas, and Vanderlei S. Bagnato.
2022. "Non-Thermal Fixed Points in Bose Gas Experiments" *Symmetry* 14, no. 4: 678.
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