k****g
发帖数: 88 | 1 【 以下文字转载自 Mathematics 讨论区 】
发信人: kuangg (Johnnie Walker), 信区: Mathematics
标 题: 小鱼行走半径问题
发信站: BBS 未名空间站 (Sun Jan 4 19:32:32 2009)
问题很简单，见笑了：
假设小鱼在2D平面以 v 自由游动，在 t 时间后它离原点的 平均 距离.
我假设它每 delta t 前进 v * delta t, 把这个运动当作 random walk, 结果2D
diffusion coefficient 是 v^2 * delta t 。这个结果显然不合理。
请高人指点。 | s**********4
发帖数: 1453 | 2 try monte carlo simulation. | s***e
发帖数: 911 | 3 = N*b^2, (b=step size)
= (t/dt)*(v*dt)^2, (dt=time per step, v=speed)
=v^2*dt*t
2d diffusion: = 4*D*t,
=> D = v^2*dt/4.
【在 k****g 的大作中提到】
: 【 以下文字转载自 Mathematics 讨论区 】
: 发信人: kuangg (Johnnie Walker), 信区: Mathematics
: 标 题: 小鱼行走半径问题
: 发信站: BBS 未名空间站 (Sun Jan 4 19:32:32 2009)
: 问题很简单，见笑了：
: 假设小鱼在2D平面以 v 自由游动，在 t 时间后它离原点的 平均 距离.
: 我假设它每 delta t 前进 v * delta t, 把这个运动当作 random walk, 结果2D
: diffusion coefficient 是 v^2 * delta t 。这个结果显然不合理。
: 请高人指点。
| i*******n
发帖数: 188 | 4 lz 算的没错呀。
这个问题需要给出一个小鱼随机改变方向的 time scale ，就是你假设的 delta t。 | k****g
发帖数: 88 | 5 realistically speaking, a fish changes its direction randomly. Therefore
there is no intrinsic time scale, or delta t.
theoretically speaking, the result depend on delta t. So when delta t -> 0,
the fish doesn't move. well, maybe this is reasonable, b/c when delta t -> 0
the fish changes direction so frequently so on average it doesn't move. | k*******a
发帖数: 772 | 6 This should be limited by uncertainty
delta(E)*delta(t)
you can think about that when delta(t) is very small, if fish changes
direction at different agnles, the acceleration will be very much different,
that means it isn't symmetric for different angles. There must be some time
scale that the motion of fish is correlated. So, there's no absolute random
motion.
,
0
【在 k****g 的大作中提到】
: realistically speaking, a fish changes its direction randomly. Therefore
: there is no intrinsic time scale, or delta t.
: theoretically speaking, the result depend on delta t. So when delta t -> 0,
: the fish doesn't move. well, maybe this is reasonable, b/c when delta t -> 0
: the fish changes direction so frequently so on average it doesn't move.
| v**g
发帖数: 20 | 7 =\int_0^T\int_0^T dtdt'
=v^2 \int_0^T\int_0^T dtdt'
\in [0,1],assume =exp{-|t-t'|/t_c},
where t_c is the correlation time of the velocity. Physically, t_c means we
have to wait such a time before we find different velocities. Then
=2*v^2*t_c*[T + t_c*exp(-T/t_c) - t_c].
As T->\infty, scales as 2*v^2*t_c*T, by definition,
the diffusion coefficient D = v^2*t_c/2. | j****c
发帖数: 19908 | 8 你没交代类似于碰撞截面的东西，就是小鱼平均游多远要改变一次方向 |
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