Relativistic hydrodynamics with phase transition

Abstract Assessing the applicability of hydrodynamic expansions close to phase transition points is crucial from either theoretical or phenomenological points of view. We explore this within the gauge/gravity duality, using the Einstein–Klein–Gordon model, a bottom-up string theory construction. This model incorporates a parameter, $$B_4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>B</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:math> , that simulates different types of phase transitions in the strongly coupled field theory existing at the boundary. We thoroughly examine the thermodynamics and dynamics of time-dependent, linearized perturbations in the spin-2, spin-1, and spin-0 sectors. Our findings suggest that ‘hydrodynamic series breakdown near transition points” is valid exclusively for second-order phase transitions, not for crossovers or first-order phase transitions. Additionally, we observe that the high-temperature and low-temperature limits of the radius of convergence for the hydrodynamic series ( $$q^2_c$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>q</mml:mi> <mml:mi>c</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math> ) are equal. We also discover that the relationship $$(\text {Max}|q^2_c|)_{\text {spin-2}}&lt; (\text {Max}|q^2_c|)_{\text {spin-0}} &lt; (\text {Max}|q^2_c|)_{\text {spin-1}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mtext>Max</mml:mtext> <mml:mo>|</mml:mo> </mml:mrow> <mml:msubsup> <mml:mi>q</mml:mi> <mml:mi>c</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> <mml:msub> <mml:mrow> <mml:mo>|</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> <mml:mtext>spin-2</mml:mtext> </mml:msub> <mml:mrow> <mml:mo>&lt;</mml:mo> <mml:mo>(</mml:mo> <mml:mtext>Max</mml:mtext> <mml:mo>|</mml:mo> </mml:mrow> <mml:msubsup> <mml:mi>q</mml:mi> <mml:mi>c</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> <mml:msub> <mml:mrow> <mml:mo>|</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> <mml:mtext>spin-0</mml:mtext> </mml:msub> <mml:mrow> <mml:mo>&lt;</mml:mo> <mml:mo>(</mml:mo> <mml:mtext>Max</mml:mtext> <mml:mo>|</mml:mo> </mml:mrow> <mml:msubsup> <mml:mi>q</mml:mi> <mml:mi>c</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> <mml:msub> <mml:mrow> <mml:mo>|</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> <mml:mtext>spin-1</mml:mtext> </mml:msub> </mml:mrow> </mml:math> is consistent for different spin sectors, regardless of the phase transition type. At the chaos point, we observe the emergence of pole-skipping behavior for both gravity and scalar perturbations at $$\omega _n = -2\pi T n i$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>ω</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> <mml:mi>π</mml:mi> <mml:mi>T</mml:mi> <mml:mi>n</mml:mi> <mml:mi>i</mml:mi> </mml:mrow> </mml:math> . Lastly, comparing the chaos momentum with $$q^2_c$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>q</mml:mi> <mml:mi>c</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math> , we find that $$q^2_{ps} &lt; q^2_c$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>q</mml:mi> <mml:mrow> <mml:mi>ps</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mo>&lt;</mml:mo> <mml:msubsup> <mml:mi>q</mml:mi> <mml:mi>c</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:mrow> </mml:math> , except for extremely high temperatures.