Relationship between the diffusion equation and the heat equation
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In physics we have the heat equation which describes the propagation of heat
$$dfrac{partial u}{partial t} = kappa dfrac{partial^2 u}{partial x^2},$$
while in biomathematics we have the diffusion equation which describes the random diffusion of a particle across space with time
$$dfrac{partial P}{partial t} = D dfrac{partial ^2 P}{partial x^2}.$$
Clearly these two equations are of similar forms, but I'm having some trouble trying to see why heat and diffusion would be related. I've always thought of heat as quite "orderly" - i.e. it propagates out in a more or less regular manner from some point of heat. However, I've always thought of diffusion as more "random" with a particle performing Brownian motion and the like to eventually reach a point ($P$ is normally related to a probability and it has a discrete recursive form, $P_{x, t+1}$).
If anyone could provide some insight, that would be great. Is this purely a coincidence or is there some basic condition which means we get analogous equations?
differential-equations pde soft-question
|
show 1 more comment
up vote
1
down vote
favorite
In physics we have the heat equation which describes the propagation of heat
$$dfrac{partial u}{partial t} = kappa dfrac{partial^2 u}{partial x^2},$$
while in biomathematics we have the diffusion equation which describes the random diffusion of a particle across space with time
$$dfrac{partial P}{partial t} = D dfrac{partial ^2 P}{partial x^2}.$$
Clearly these two equations are of similar forms, but I'm having some trouble trying to see why heat and diffusion would be related. I've always thought of heat as quite "orderly" - i.e. it propagates out in a more or less regular manner from some point of heat. However, I've always thought of diffusion as more "random" with a particle performing Brownian motion and the like to eventually reach a point ($P$ is normally related to a probability and it has a discrete recursive form, $P_{x, t+1}$).
If anyone could provide some insight, that would be great. Is this purely a coincidence or is there some basic condition which means we get analogous equations?
differential-equations pde soft-question
1
These literally are the same thing. Heat is the diffusion of thermal energy, which is random in nature (e.g. the motion of every particle in a medium in which heat is propagating is random).
– Arturo don Juan
Nov 7 '14 at 1:23
Isn't $kappa$ a constant but $D$ a function?
– lhf
Nov 7 '14 at 1:24
$D$ is supposed to depend on the medium of diffusion, so it can be regarded as a constant for each particular medium, but we also have "fractional diffusion" and the like where $D=D(t) = t^{-1/2}$ for example. But $kappa$ is similarly defined and is characteristic of the medium propagating heat as well... So I guess they could potentially both be regarded as functions of the properties of the media?
– jamesh625
Nov 7 '14 at 1:30
@ArturoDonJuan But then why is the probability for a particle's position of the same form as the equation that governs its motion? I'm just surprised that they're exactly the same and not even slightly different. Is the temperature supposed to be analogous to the probability of a bit of heat-energy being there?
– jamesh625
Nov 7 '14 at 1:33
The diffusion equation tells you how the probability density function of a Wiener process evolves over time. It's probably obvious why this also describes evolution of temperature if you know the definition of temperature. But mathematically they are the exact same differential equation.
– mathematician
Nov 7 '14 at 1:42
|
show 1 more comment
up vote
1
down vote
favorite
up vote
1
down vote
favorite
In physics we have the heat equation which describes the propagation of heat
$$dfrac{partial u}{partial t} = kappa dfrac{partial^2 u}{partial x^2},$$
while in biomathematics we have the diffusion equation which describes the random diffusion of a particle across space with time
$$dfrac{partial P}{partial t} = D dfrac{partial ^2 P}{partial x^2}.$$
Clearly these two equations are of similar forms, but I'm having some trouble trying to see why heat and diffusion would be related. I've always thought of heat as quite "orderly" - i.e. it propagates out in a more or less regular manner from some point of heat. However, I've always thought of diffusion as more "random" with a particle performing Brownian motion and the like to eventually reach a point ($P$ is normally related to a probability and it has a discrete recursive form, $P_{x, t+1}$).
If anyone could provide some insight, that would be great. Is this purely a coincidence or is there some basic condition which means we get analogous equations?
differential-equations pde soft-question
In physics we have the heat equation which describes the propagation of heat
$$dfrac{partial u}{partial t} = kappa dfrac{partial^2 u}{partial x^2},$$
while in biomathematics we have the diffusion equation which describes the random diffusion of a particle across space with time
$$dfrac{partial P}{partial t} = D dfrac{partial ^2 P}{partial x^2}.$$
Clearly these two equations are of similar forms, but I'm having some trouble trying to see why heat and diffusion would be related. I've always thought of heat as quite "orderly" - i.e. it propagates out in a more or less regular manner from some point of heat. However, I've always thought of diffusion as more "random" with a particle performing Brownian motion and the like to eventually reach a point ($P$ is normally related to a probability and it has a discrete recursive form, $P_{x, t+1}$).
If anyone could provide some insight, that would be great. Is this purely a coincidence or is there some basic condition which means we get analogous equations?
differential-equations pde soft-question
differential-equations pde soft-question
asked Nov 7 '14 at 1:11
jamesh625
5661417
5661417
1
These literally are the same thing. Heat is the diffusion of thermal energy, which is random in nature (e.g. the motion of every particle in a medium in which heat is propagating is random).
– Arturo don Juan
Nov 7 '14 at 1:23
Isn't $kappa$ a constant but $D$ a function?
– lhf
Nov 7 '14 at 1:24
$D$ is supposed to depend on the medium of diffusion, so it can be regarded as a constant for each particular medium, but we also have "fractional diffusion" and the like where $D=D(t) = t^{-1/2}$ for example. But $kappa$ is similarly defined and is characteristic of the medium propagating heat as well... So I guess they could potentially both be regarded as functions of the properties of the media?
– jamesh625
Nov 7 '14 at 1:30
@ArturoDonJuan But then why is the probability for a particle's position of the same form as the equation that governs its motion? I'm just surprised that they're exactly the same and not even slightly different. Is the temperature supposed to be analogous to the probability of a bit of heat-energy being there?
– jamesh625
Nov 7 '14 at 1:33
The diffusion equation tells you how the probability density function of a Wiener process evolves over time. It's probably obvious why this also describes evolution of temperature if you know the definition of temperature. But mathematically they are the exact same differential equation.
– mathematician
Nov 7 '14 at 1:42
|
show 1 more comment
1
These literally are the same thing. Heat is the diffusion of thermal energy, which is random in nature (e.g. the motion of every particle in a medium in which heat is propagating is random).
– Arturo don Juan
Nov 7 '14 at 1:23
Isn't $kappa$ a constant but $D$ a function?
– lhf
Nov 7 '14 at 1:24
$D$ is supposed to depend on the medium of diffusion, so it can be regarded as a constant for each particular medium, but we also have "fractional diffusion" and the like where $D=D(t) = t^{-1/2}$ for example. But $kappa$ is similarly defined and is characteristic of the medium propagating heat as well... So I guess they could potentially both be regarded as functions of the properties of the media?
– jamesh625
Nov 7 '14 at 1:30
@ArturoDonJuan But then why is the probability for a particle's position of the same form as the equation that governs its motion? I'm just surprised that they're exactly the same and not even slightly different. Is the temperature supposed to be analogous to the probability of a bit of heat-energy being there?
– jamesh625
Nov 7 '14 at 1:33
The diffusion equation tells you how the probability density function of a Wiener process evolves over time. It's probably obvious why this also describes evolution of temperature if you know the definition of temperature. But mathematically they are the exact same differential equation.
– mathematician
Nov 7 '14 at 1:42
1
1
These literally are the same thing. Heat is the diffusion of thermal energy, which is random in nature (e.g. the motion of every particle in a medium in which heat is propagating is random).
– Arturo don Juan
Nov 7 '14 at 1:23
These literally are the same thing. Heat is the diffusion of thermal energy, which is random in nature (e.g. the motion of every particle in a medium in which heat is propagating is random).
– Arturo don Juan
Nov 7 '14 at 1:23
Isn't $kappa$ a constant but $D$ a function?
– lhf
Nov 7 '14 at 1:24
Isn't $kappa$ a constant but $D$ a function?
– lhf
Nov 7 '14 at 1:24
$D$ is supposed to depend on the medium of diffusion, so it can be regarded as a constant for each particular medium, but we also have "fractional diffusion" and the like where $D=D(t) = t^{-1/2}$ for example. But $kappa$ is similarly defined and is characteristic of the medium propagating heat as well... So I guess they could potentially both be regarded as functions of the properties of the media?
– jamesh625
Nov 7 '14 at 1:30
$D$ is supposed to depend on the medium of diffusion, so it can be regarded as a constant for each particular medium, but we also have "fractional diffusion" and the like where $D=D(t) = t^{-1/2}$ for example. But $kappa$ is similarly defined and is characteristic of the medium propagating heat as well... So I guess they could potentially both be regarded as functions of the properties of the media?
– jamesh625
Nov 7 '14 at 1:30
@ArturoDonJuan But then why is the probability for a particle's position of the same form as the equation that governs its motion? I'm just surprised that they're exactly the same and not even slightly different. Is the temperature supposed to be analogous to the probability of a bit of heat-energy being there?
– jamesh625
Nov 7 '14 at 1:33
@ArturoDonJuan But then why is the probability for a particle's position of the same form as the equation that governs its motion? I'm just surprised that they're exactly the same and not even slightly different. Is the temperature supposed to be analogous to the probability of a bit of heat-energy being there?
– jamesh625
Nov 7 '14 at 1:33
The diffusion equation tells you how the probability density function of a Wiener process evolves over time. It's probably obvious why this also describes evolution of temperature if you know the definition of temperature. But mathematically they are the exact same differential equation.
– mathematician
Nov 7 '14 at 1:42
The diffusion equation tells you how the probability density function of a Wiener process evolves over time. It's probably obvious why this also describes evolution of temperature if you know the definition of temperature. But mathematically they are the exact same differential equation.
– mathematician
Nov 7 '14 at 1:42
|
show 1 more comment
1 Answer
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I'll try to answer your question better here.
In the heat equation, what is the base assumption? It is that the heat flow $frac{partial u}{partial t}$ is proportional to the second spacial-derivative (laplacian) of the temperature at the point.
$$frac{partial u}{partial t}proptonabla^2 u$$
In the diffusion equation with the diffusivity $D$ being constant, what is the base assumption? It is that the diffusion (rate of change of the density of particles with respect to time) of the particles at a point is proportional to the second spacial-derivative (laplacian) of the density of those particles at that point.
$$frac{partial P}{partial t}proptonabla^2 P$$
So how are they related? Well, they literally revolve around the same base assumption - that the amount of whatever they're dealing with (e.g. temperature or density) is conserved. Thus, both of these are applications of the Continuity Equation.
If we have a density of "something" $H$ in space, which has a local current density $mathbf{J}$ (i.e. amount of $H$ passing through a sufficiently small volume per unit time, divided by that volume), then the assumption that "no $H$ can be created or destroyed" is equivalent to the continuity equation,
$$frac{partial H}{partial t} + nablacdot mathbf{J} = 0$$
If we would like to make contact with the heat equation, we define $H$ to be local measure of thermal energy (i.e. temperature - we assume thermal energy can neither be created nor destroyed, but only transferred from one place to another), and make use of Fick's law of diffusion, which states that the current density of a diffusive material/stuff is proportional to the negative of its gradient.
$$mathbf{J}propto -nabla H$$
Letting the proportionality constant be equal to $D$, we get the heat equation.
$$frac{partial H}{partial t}=Dnabla^2H$$
This is really just the steady-state diffusion equation, which can be generalized to any situation which revolves around the same basic assumptions. An example of this is Schrodinger's equation for a free-particle.
$$frac{partial psi}{partial t}=inabla^2 psi$$
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1 Answer
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1 Answer
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active
oldest
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active
oldest
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active
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up vote
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I'll try to answer your question better here.
In the heat equation, what is the base assumption? It is that the heat flow $frac{partial u}{partial t}$ is proportional to the second spacial-derivative (laplacian) of the temperature at the point.
$$frac{partial u}{partial t}proptonabla^2 u$$
In the diffusion equation with the diffusivity $D$ being constant, what is the base assumption? It is that the diffusion (rate of change of the density of particles with respect to time) of the particles at a point is proportional to the second spacial-derivative (laplacian) of the density of those particles at that point.
$$frac{partial P}{partial t}proptonabla^2 P$$
So how are they related? Well, they literally revolve around the same base assumption - that the amount of whatever they're dealing with (e.g. temperature or density) is conserved. Thus, both of these are applications of the Continuity Equation.
If we have a density of "something" $H$ in space, which has a local current density $mathbf{J}$ (i.e. amount of $H$ passing through a sufficiently small volume per unit time, divided by that volume), then the assumption that "no $H$ can be created or destroyed" is equivalent to the continuity equation,
$$frac{partial H}{partial t} + nablacdot mathbf{J} = 0$$
If we would like to make contact with the heat equation, we define $H$ to be local measure of thermal energy (i.e. temperature - we assume thermal energy can neither be created nor destroyed, but only transferred from one place to another), and make use of Fick's law of diffusion, which states that the current density of a diffusive material/stuff is proportional to the negative of its gradient.
$$mathbf{J}propto -nabla H$$
Letting the proportionality constant be equal to $D$, we get the heat equation.
$$frac{partial H}{partial t}=Dnabla^2H$$
This is really just the steady-state diffusion equation, which can be generalized to any situation which revolves around the same basic assumptions. An example of this is Schrodinger's equation for a free-particle.
$$frac{partial psi}{partial t}=inabla^2 psi$$
add a comment |
up vote
4
down vote
accepted
I'll try to answer your question better here.
In the heat equation, what is the base assumption? It is that the heat flow $frac{partial u}{partial t}$ is proportional to the second spacial-derivative (laplacian) of the temperature at the point.
$$frac{partial u}{partial t}proptonabla^2 u$$
In the diffusion equation with the diffusivity $D$ being constant, what is the base assumption? It is that the diffusion (rate of change of the density of particles with respect to time) of the particles at a point is proportional to the second spacial-derivative (laplacian) of the density of those particles at that point.
$$frac{partial P}{partial t}proptonabla^2 P$$
So how are they related? Well, they literally revolve around the same base assumption - that the amount of whatever they're dealing with (e.g. temperature or density) is conserved. Thus, both of these are applications of the Continuity Equation.
If we have a density of "something" $H$ in space, which has a local current density $mathbf{J}$ (i.e. amount of $H$ passing through a sufficiently small volume per unit time, divided by that volume), then the assumption that "no $H$ can be created or destroyed" is equivalent to the continuity equation,
$$frac{partial H}{partial t} + nablacdot mathbf{J} = 0$$
If we would like to make contact with the heat equation, we define $H$ to be local measure of thermal energy (i.e. temperature - we assume thermal energy can neither be created nor destroyed, but only transferred from one place to another), and make use of Fick's law of diffusion, which states that the current density of a diffusive material/stuff is proportional to the negative of its gradient.
$$mathbf{J}propto -nabla H$$
Letting the proportionality constant be equal to $D$, we get the heat equation.
$$frac{partial H}{partial t}=Dnabla^2H$$
This is really just the steady-state diffusion equation, which can be generalized to any situation which revolves around the same basic assumptions. An example of this is Schrodinger's equation for a free-particle.
$$frac{partial psi}{partial t}=inabla^2 psi$$
add a comment |
up vote
4
down vote
accepted
up vote
4
down vote
accepted
I'll try to answer your question better here.
In the heat equation, what is the base assumption? It is that the heat flow $frac{partial u}{partial t}$ is proportional to the second spacial-derivative (laplacian) of the temperature at the point.
$$frac{partial u}{partial t}proptonabla^2 u$$
In the diffusion equation with the diffusivity $D$ being constant, what is the base assumption? It is that the diffusion (rate of change of the density of particles with respect to time) of the particles at a point is proportional to the second spacial-derivative (laplacian) of the density of those particles at that point.
$$frac{partial P}{partial t}proptonabla^2 P$$
So how are they related? Well, they literally revolve around the same base assumption - that the amount of whatever they're dealing with (e.g. temperature or density) is conserved. Thus, both of these are applications of the Continuity Equation.
If we have a density of "something" $H$ in space, which has a local current density $mathbf{J}$ (i.e. amount of $H$ passing through a sufficiently small volume per unit time, divided by that volume), then the assumption that "no $H$ can be created or destroyed" is equivalent to the continuity equation,
$$frac{partial H}{partial t} + nablacdot mathbf{J} = 0$$
If we would like to make contact with the heat equation, we define $H$ to be local measure of thermal energy (i.e. temperature - we assume thermal energy can neither be created nor destroyed, but only transferred from one place to another), and make use of Fick's law of diffusion, which states that the current density of a diffusive material/stuff is proportional to the negative of its gradient.
$$mathbf{J}propto -nabla H$$
Letting the proportionality constant be equal to $D$, we get the heat equation.
$$frac{partial H}{partial t}=Dnabla^2H$$
This is really just the steady-state diffusion equation, which can be generalized to any situation which revolves around the same basic assumptions. An example of this is Schrodinger's equation for a free-particle.
$$frac{partial psi}{partial t}=inabla^2 psi$$
I'll try to answer your question better here.
In the heat equation, what is the base assumption? It is that the heat flow $frac{partial u}{partial t}$ is proportional to the second spacial-derivative (laplacian) of the temperature at the point.
$$frac{partial u}{partial t}proptonabla^2 u$$
In the diffusion equation with the diffusivity $D$ being constant, what is the base assumption? It is that the diffusion (rate of change of the density of particles with respect to time) of the particles at a point is proportional to the second spacial-derivative (laplacian) of the density of those particles at that point.
$$frac{partial P}{partial t}proptonabla^2 P$$
So how are they related? Well, they literally revolve around the same base assumption - that the amount of whatever they're dealing with (e.g. temperature or density) is conserved. Thus, both of these are applications of the Continuity Equation.
If we have a density of "something" $H$ in space, which has a local current density $mathbf{J}$ (i.e. amount of $H$ passing through a sufficiently small volume per unit time, divided by that volume), then the assumption that "no $H$ can be created or destroyed" is equivalent to the continuity equation,
$$frac{partial H}{partial t} + nablacdot mathbf{J} = 0$$
If we would like to make contact with the heat equation, we define $H$ to be local measure of thermal energy (i.e. temperature - we assume thermal energy can neither be created nor destroyed, but only transferred from one place to another), and make use of Fick's law of diffusion, which states that the current density of a diffusive material/stuff is proportional to the negative of its gradient.
$$mathbf{J}propto -nabla H$$
Letting the proportionality constant be equal to $D$, we get the heat equation.
$$frac{partial H}{partial t}=Dnabla^2H$$
This is really just the steady-state diffusion equation, which can be generalized to any situation which revolves around the same basic assumptions. An example of this is Schrodinger's equation for a free-particle.
$$frac{partial psi}{partial t}=inabla^2 psi$$
edited Nov 23 at 19:31
answered Nov 7 '14 at 2:27
Arturo don Juan
1,8061031
1,8061031
add a comment |
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These literally are the same thing. Heat is the diffusion of thermal energy, which is random in nature (e.g. the motion of every particle in a medium in which heat is propagating is random).
– Arturo don Juan
Nov 7 '14 at 1:23
Isn't $kappa$ a constant but $D$ a function?
– lhf
Nov 7 '14 at 1:24
$D$ is supposed to depend on the medium of diffusion, so it can be regarded as a constant for each particular medium, but we also have "fractional diffusion" and the like where $D=D(t) = t^{-1/2}$ for example. But $kappa$ is similarly defined and is characteristic of the medium propagating heat as well... So I guess they could potentially both be regarded as functions of the properties of the media?
– jamesh625
Nov 7 '14 at 1:30
@ArturoDonJuan But then why is the probability for a particle's position of the same form as the equation that governs its motion? I'm just surprised that they're exactly the same and not even slightly different. Is the temperature supposed to be analogous to the probability of a bit of heat-energy being there?
– jamesh625
Nov 7 '14 at 1:33
The diffusion equation tells you how the probability density function of a Wiener process evolves over time. It's probably obvious why this also describes evolution of temperature if you know the definition of temperature. But mathematically they are the exact same differential equation.
– mathematician
Nov 7 '14 at 1:42