Anomalous velocity distributions in slow quantum-tunneling chemical reactions

Recent work [R. Wild ] has provided an experimental break-through in the realization of a quantum-tunneling reaction involving a proton transfer. The reaction <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:msup><a:mi>D</a:mi><a:mo>−</a:mo></a:msup><a:mo>+</a:mo><a:msub><a:mi>H</a:mi><a:mn>2</a:mn></a:msub><a:mo>→</a:mo><a:msup><a:mi>H</a:mi><a:mo>−</a:mo></a:msup><a:mo>+</a:mo><a:mi>H</a:mi><a:mi>D</a:mi></a:mrow></a:math> has an extremely slow reaction rate as it can happen only via quantum tunneling, thus requiring an extremely large density of the reactants in the ion trap. At these high densities strong deviations from Maxwell-Boltzmann statistics are observed. Here we develop a consistent generalized statistical mechanics theory for the above nonequilibrium situation involving quantum effects at high densities. The trapped ions are treated in a superstatistical way and a <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi>q</b:mi></b:math>-Maxwellian velocity distribution with a universal dependence of the entropic index <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi>q</c:mi></c:math> on the density <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mi>n</d:mi></d:math> of the buffer gas is derived. We show that the velocity distribution of the ions is non-Maxwellian, more precisely <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mi>q</e:mi></e:math>-Gaussian, i.e., <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mrow><f:mi>p</f:mi><f:mrow><f:mo>(</f:mo><f:mi>v</f:mi><f:mo>)</f:mo></f:mrow><f:mo>∝</f:mo><f:msup><f:mi>v</f:mi><f:mn>2</f:mn></f:msup><f:msup><f:mrow><f:mo>[</f:mo><f:mn>1</f:mn><f:mo>+</f:mo><f:mrow><f:mo>(</f:mo><f:mi>q</f:mi><f:mo>−</f:mo><f:mn>1</f:mn><f:mo>)</f:mo></f:mrow><f:mover accent="true"><f:mi>β</f:mi><f:mo>̃</f:mo></f:mover><f:msup><f:mi>v</f:mi><f:mn>2</f:mn></f:msup><f:mo>]</f:mo></f:mrow><f:mrow><f:mn>1</f:mn><f:mo>/</f:mo><f:mo>(</f:mo><f:mn>1</f:mn><f:mo>−</f:mo><f:mi>q</f:mi><f:mo>)</f:mo></f:mrow></f:msup></f:mrow></f:math>, with entropic index <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mrow><h:mi>q</h:mi><h:mo>&gt;</h:mo><h:mn>1</h:mn></h:mrow></h:math> depending on the density <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:mi>n</i:mi></i:math> of <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:msub><j:mi>H</j:mi><j:mn>2</j:mn></j:msub></j:math> molecules, in excellent agreement with the experimental observations of Wild Our theory also makes predictions on the statistics of temperature fluctuations in the ion trap which can be tested in future experiments. Through the superstatistical approach, we obtain an analytical expression for <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"><k:mrow><k:mi>q</k:mi><k:mo>(</k:mo><k:mi>n</k:mi><k:mo>)</k:mo></k:mrow></k:math> which is consistent with the available experimental data, and which yields <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"><l:mrow><l:msub><l:mo movablelimits="true" form="prefix">lim</l:mo><l:mrow><l:mi>n</l:mi><l:mo>→</l:mo><l:mn>0</l:mn></l:mrow></l:msub><l:mi>q</l:mi><l:mrow><l:mo>(</l:mo><l:mi>n</l:mi><l:mo>)</l:mo></l:mrow><l:mo>=</l:mo><l:mn>1</l:mn></l:mrow></l:math>, i.e., recovering the Maxwell-Boltzmann distribution in the ideal gas limit, as well as <o:math xmlns:o="http://www.w3.org/1998/Math/MathML"><o:mrow><o:msub><o:mo movablelimits="true" form="prefix">lim</o:mo><o:mrow><o:mi>n</o:mi><o:mo>→</o:mo><o:mi>∞</o:mi></o:mrow></o:msub><o:mi>q</o:mi><o:mrow><o:mo>(</o:mo><o:mi>n</o:mi><o:mo>)</o:mo></o:mrow><o:mo>=</o:mo><o:mn>7</o:mn><o:mo>/</o:mo><o:mn>5</o:mn></o:mrow></o:math>.