The persistence exponent <img src="Edit_8589f062-08af-48bf-9fa4-ea64e4f98789.png" alt="" /> for the simple diffusion equation <img src="Edit_8bd8b3b8-7f1f-4ea5-a5f5-c5ccc20288f4.png&q...The persistence exponent <img src="Edit_8589f062-08af-48bf-9fa4-ea64e4f98789.png" alt="" /> for the simple diffusion equation <img src="Edit_8bd8b3b8-7f1f-4ea5-a5f5-c5ccc20288f4.png" alt="" /> , with random Gaussian initial condition, has been calculated exactly using a method known as selective averaging. The probability that the value of the field <img src="Edit_cc47d602-457a-4e52-93d8-acc18dcaf933.png" alt="" /> at a specified spatial coordinate remains positive throughout for a certain time<em> t</em> behaves as <img src="Edit_aacdd656-f2c2-4cde-ba3c-1b32bf053b3b.png" alt="" /> for asymptotically large time <em>t</em>. The value of <img src="Edit_77272c69-2a19-4918-a183-7db96b262c7a.png" alt="" /> , calculated here for any integer dimension <em>d</em>, is <img src="Edit_bc64e52a-d6d0-4b63-8ef3-aa0f9d3c39cc.png" alt="" /> for <img src="Edit_becf7ae7-0ae4-43a6-9a41-017f25747517.png" alt="" /> and 1 otherwise. This exact theoretical result is being reported possibly for the first time and is not in agreement with the accepted values <img src="Edit_fbefbfcf-d76b-4eeb-a5f5-d8afda4a1a0c.png" alt="" /> for <img src="Edit_ec927d57-c273-40dd-8126-706443b57534.png" alt="" /> respectively.展开更多
文摘The persistence exponent <img src="Edit_8589f062-08af-48bf-9fa4-ea64e4f98789.png" alt="" /> for the simple diffusion equation <img src="Edit_8bd8b3b8-7f1f-4ea5-a5f5-c5ccc20288f4.png" alt="" /> , with random Gaussian initial condition, has been calculated exactly using a method known as selective averaging. The probability that the value of the field <img src="Edit_cc47d602-457a-4e52-93d8-acc18dcaf933.png" alt="" /> at a specified spatial coordinate remains positive throughout for a certain time<em> t</em> behaves as <img src="Edit_aacdd656-f2c2-4cde-ba3c-1b32bf053b3b.png" alt="" /> for asymptotically large time <em>t</em>. The value of <img src="Edit_77272c69-2a19-4918-a183-7db96b262c7a.png" alt="" /> , calculated here for any integer dimension <em>d</em>, is <img src="Edit_bc64e52a-d6d0-4b63-8ef3-aa0f9d3c39cc.png" alt="" /> for <img src="Edit_becf7ae7-0ae4-43a6-9a41-017f25747517.png" alt="" /> and 1 otherwise. This exact theoretical result is being reported possibly for the first time and is not in agreement with the accepted values <img src="Edit_fbefbfcf-d76b-4eeb-a5f5-d8afda4a1a0c.png" alt="" /> for <img src="Edit_ec927d57-c273-40dd-8126-706443b57534.png" alt="" /> respectively.
