<p>Based on well-known heuristic arguments that provide an upper bound for the relaxation time of zero-range processes and trying to establish a connection between the results obtained in Landim et al. (Ann Probab 24(4), 1996), Morris (Ann Probab 34(5):1645–1664, 2006) and Nagahata (Stoch Process Appl 120(6):949–958, 2010) we proposed to study the zero-range process associated with the following family of rate functions: <Equation ID="Equ15"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="40324_2025_389_Article_Equ15.gif" Format="GIF" Height="54" Rendition="HTML" Resolution="72" Type="Linedraw" Width="303" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} g_{\alpha }(k) = {\left\{ \begin{array}{ll} \frac{(k+1)^{1-\alpha }-1}{1-\alpha } & \text {if}\ \alpha \in [0,1) \cup (1,\infty ) \\ \ln (k+1) & \text {if}\ \alpha =1. \end{array}\right. } \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>g</mi> <mi>α</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd columnalign="left"> <mfrac> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>1</mn> <mo>-</mo> <mi>α</mi> </mrow> </msup> <mo>-</mo> <mn>1</mn> </mrow> <mrow> <mn>1</mn> <mo>-</mo> <mi>α</mi> </mrow> </mfrac> </mtd> <mtd columnalign="left"> <mrow> <mtext>if</mtext> <mspace width="4pt" /> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>∪</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <mo>ln</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mtext>if</mtext> <mspace width="4pt" /> <mi>α</mi> <mo>=</mo> <mn>1</mn> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>We state a conjecture about the order of the relaxation time for those processes and provide numerical evidence throughout deterministic iterative methods and Monte-Carlo simulations.</p>

Relaxation time for a parametric family of zero-range processes

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2025年5月5日

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F. Hernández-Romero,
M. Jara,
F. Valentim

摘要

Based on well-known heuristic arguments that provide an upper bound for the relaxation time of zero-range processes and trying to establish a connection between the results obtained in Landim et al. (Ann Probab 24(4), 1996), Morris (Ann Probab 34(5):1645–1664, 2006) and Nagahata (Stoch Process Appl 120(6):949–958, 2010) we proposed to study the zero-range process associated with the following family of rate functions: $\begin{aligned} g_{\alpha }(k) = {\left\{ \begin{array}{ll} \frac{(k+1)^{1-\alpha }-1}{1-\alpha } & \text {if}\ \alpha \in [0,1) \cup (1,\infty ) \\ \ln (k+1) & \text {if}\ \alpha =1. \end{array}\right. } \end{aligned}$ $\begin{matrix} g_{α} (k) = \{\begin{matrix} \frac{{(k + 1)}^{1 - α} - 1}{1 - α} & if α \in [0, 1) \cup (1, \infty) \\ ln (k + 1) & if α = 1 . \end{matrix}) \end{matrix}$ We state a conjecture about the order of the relaxation time for those processes and provide numerical evidence throughout deterministic iterative methods and Monte-Carlo simulations.