Langevin equation - Revision history

Jola: Reverted edits by YlzEnr (Talk); changed back to last version by Salva

2007-06-15T14:30:30Z

Reverted edits by YlzEnr (Talk); changed back to last version by Salva

← Older revision		Revision as of 14:30, 15 June 2007
Line 2:		Line 2:
	the Langevin equation is		the Langevin equation is
	:<math>		:<math>
-	dU(t) = - U(t) \frac{dt}{\tau} \frac{2 u'}{\tau}^{1/2} dW(t)	+	dU(t) = - U(t) \frac{dt}{\tau} + \frac{2 u'}{\tau}^{1/2} dW(t)
	</math>		</math>

Line 11:		Line 11:

	:<math>		:<math>
-	U(t \Delta t) = U(t) - U(t) \frac{\Delta t}{\tau} \frac{2 u' \Delta t}{\tau}^{1/2} \mathcal{N}	+	U(t+\Delta t) = U(t) - U(t) \frac{\Delta t}{\tau} + \frac{2 u' \Delta t}{\tau}^{1/2} \mathcal{N}
	</math>		</math>

YlzEnr at 12:20, 15 June 2007

2007-06-15T12:20:12Z

← Older revision		Revision as of 12:20, 15 June 2007
Line 2:		Line 2:
	the Langevin equation is		the Langevin equation is
	:<math>		:<math>
-	dU(t) = - U(t) \frac{dt}{\tau} + \frac{2 u'}{\tau}^{1/2} dW(t)	+	dU(t) = - U(t) \frac{dt}{\tau} \frac{2 u'}{\tau}^{1/2} dW(t)
	</math>		</math>

Line 11:		Line 11:

	:<math>		:<math>
-	U(t+\Delta t) = U(t) - U(t) \frac{\Delta t}{\tau} + \frac{2 u' \Delta t}{\tau}^{1/2} \mathcal{N}	+	U(t \Delta t) = U(t) - U(t) \frac{\Delta t}{\tau} \frac{2 u' \Delta t}{\tau}^{1/2} \mathcal{N}
	</math>		</math>

Salva at 09:29, 14 November 2005

2005-11-14T09:29:53Z

← Older revision		Revision as of 09:29, 14 November 2005
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	The Wiener process can be understood as Gaussian random variable with 0 mean		The Wiener process can be understood as Gaussian random variable with 0 mean
	and variance <math> dt</math>		and variance <math> dt</math>
		+
		+	{{Stub}}

Salva at 17:30, 2 November 2005

2005-11-02T17:30:45Z

← Older revision		Revision as of 17:30, 2 November 2005
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	where <math> dW(t) </math> is a Wiener process.		where <math> dW(t) </math> is a Wiener process.
-	<math> u' </math> is the turbulence intensity and math> \tau </math> ia Lagrangian time-scale.	+	<math> u' </math> is the turbulence intensity and <math> \tau </math> a Lagrangian time-scale.

	Th finite difference approximation of the above equation is		Th finite difference approximation of the above equation is

Salva at 17:29, 2 November 2005

2005-11-02T17:29:42Z

New page

The stochastic differential equation (SDE) for velocity component <math> U(t) </math>,
the Langevin equation is
:<math>
dU(t) = - U(t) \frac{dt}{\tau} + \frac{2 u'}{\tau}^{1/2} dW(t)
</math>

where <math> dW(t) </math> is a Wiener process.
<math> u' </math> is the turbulence intensity and math> \tau </math> ia Lagrangian time-scale.

Th finite difference approximation of the above equation is

:<math>
U(t+\Delta t) = U(t) - U(t) \frac{\Delta t}{\tau} + \frac{2 u' \Delta t}{\tau}^{1/2} \mathcal{N}
</math>

where <math> \mathcal{N} </math> is a standardized Gaussian random variable with 0 mean an unity variance
which is independent of <math> U </math> on all other time steps (Pope 1994).
The Wiener process can be understood as Gaussian random variable with 0 mean
and variance <math> dt</math>