Hyperuniformity in mass transport processes with center-of-mass conservation: some exact results

Abstract We characterize the steady-state static and dynamic properties in a broad class of mass transport processes on a periodic hypercubic lattice of volume L d , where both the mass and center-of-mass (CoM) remain conserved and the detailed balance is violated in the bulk; we specifically consider the models in the d = 1 and 2 dimensions. Using a microscopic approach, we exactly determine the decay (or growth) exponents for various dynamic and static correlation functions. We show that despite constrained dynamics due to CoM conservation (CoMC), the density relaxation is indeed diffusive. However, the fluctuation properties are strikingly different from those in diffusive systems with a single (mass) conservation law as dynamic and static fluctuations are more suppressed in systems with CoMC, resulting in an extreme (‘class-I’) hyperuniformity in certain cases. In the thermodynamic limit, the steady-state variance <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo fence="false" stretchy="false">⟨</mml:mo> <mml:msup> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">Q</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:msub> <mml:mo fence="false" stretchy="false">⟩</mml:mo> <mml:mi>c</mml:mi> </mml:msub> </mml:mrow> </mml:math> of time-integrated bond current <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">Q</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> across a bond in the time interval T exhibits the following long-time behavior: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo fence="false" stretchy="false">⟨</mml:mo> <mml:msup> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">Q</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:msub> <mml:mo fence="false" stretchy="false">⟩</mml:mo> <mml:mi>c</mml:mi> </mml:msub> <mml:mo>≃</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mi>T</mml:mi> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:msup> <mml:mi>T</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mi>d</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> . The exponents governing the small-frequency behavior of the power spectra <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>J</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>≃</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:mrow> <mml:mrow> <mml:mi>Const</mml:mi> <mml:mo>.</mml:mo> </mml:mrow> </mml:mrow> <mml:msup> <mml:mi>f</mml:mi> <mml:mrow> <mml:msub> <mml:mi>ψ</mml:mi> <mml:mi>J</mml:mi> </mml:msub> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> for bond current are exactly determined as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>ψ</mml:mi> <mml:mi>J</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> and 2 in d = 1 and 2 dimensions, respectively; the corresponding unequal-time current-current correlation function decay as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mi>t</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>5</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> and t −3 as a function of time t in d = 1 and 2 dimensions, respectively. Remarkably, depending on the dimensions and microscopic details, the prefactor A 1 can vanish, causing the variance to eventually saturate , implying a class-I ‘dynamic hyperuniformity’. We also compute the static structure factor S ( q ), which varies as the square of the wave number q in the small- q limit, i.e. <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>∼</mml:mo> <mml:msup> <mml:mi>q</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math>