3.3.1 Self-diffusion and random walk

A particle initially located at r<sup>0</sup> has equal probabilities of 1/6 to move in ð Þ �x; �y; �z directions. For mathematical simplicity, we restrict ourselves to 1D random walk of a dizzy individual, who moves to the right or to the left with a 50:50 chance. Initially (at time t ¼ 0), the individual is located at x<sup>0</sup> ¼ 0 and starts moving in a direction represented by Δx ¼ �l where þl and �l indicate the right and left distances that the individual travels with an equal probability, respectively. At the next step, t<sup>1</sup> ¼ t<sup>0</sup> þ Δt ¼ Δt, the individual's location is found at

$$
\Delta \mathbf{x}\_1 = \mathbf{x}\_0 + \Delta \mathbf{x}\_1 = \Delta \mathbf{x}\_1 \tag{19}
$$

where Δx<sup>1</sup> can be þl or �l. At the time of the second step, t<sup>2</sup> ¼ t<sup>1</sup> þ Δt ¼ 2Δt, the position is

$$
\omega\_2 = \omega\_1 + \Delta\mathfrak{x}\_2 = \Delta\mathfrak{x}\_1 + \Delta\mathfrak{x}\_2 \tag{20}
$$

where Δx<sup>2</sup> ¼ �l. At tn ¼ nΔt (n ≫ 1), the position may be expressed as

$$\mathbf{x}\_{n} = \Delta \mathbf{x}\_{1} + \Delta \mathbf{x}\_{2} + \dots + \Delta \mathbf{x}\_{n-1} + \Delta \mathbf{x}\_{n} = \sum\_{i=1}^{n} \Delta \mathbf{x}\_{i} \tag{21}$$

If there are a number of dizzy individuals and we can determine an average for their seemingly random movements, then

$$
\langle \mathbf{x}\_n \rangle = \sum\_{i=1}^n \langle \Delta \mathbf{x}\_i \rangle = n \langle \Delta \mathbf{x} \rangle = \mathbf{0} \tag{22}
$$

because Δx has a 50:50 chance of þl and �l:

$$
\langle \Delta \mathbf{x} \rangle = (+l)\frac{\mathbf{1}}{2} + (-l)\frac{\mathbf{1}}{2} = \mathbf{0} \tag{23}
$$

Now let us calculate a mean of x2:

$$\begin{aligned} \left< \mathbf{x}\_n^2 \right> &= \left< (\Delta \mathbf{x}\_1 + \dots \Delta \mathbf{x}\_n) \cdot (\Delta \mathbf{x}\_1 + \dots \Delta \mathbf{x}\_n) \right> \\ &= \left< \Delta \mathbf{x}\_1^2 + \Delta \mathbf{x}\_1 \cdot \Delta \mathbf{x}\_2 + \dots + \Delta \mathbf{x}\_1 \cdot \Delta \mathbf{x}\_n \right. \\ &\quad + \Delta \mathbf{x}\_2 \cdot \Delta \mathbf{x}\_1 + \Delta \mathbf{x}\_2^2 + \dots + \Delta \mathbf{x}\_2 \cdot \Delta \mathbf{x}\_n \\ &\quad + \dotsb \\ &\quad + \Delta \mathbf{x}\_n \cdot \Delta \mathbf{x}\_1 + \Delta \mathbf{x}\_n \cdot \Delta \mathbf{x}\_2 + \dotsb + \Delta \mathbf{x}\_n^2 \right> \end{aligned} \tag{24}$$

C xð Þ¼ � y; t C xð Þ� ; t

Fundamentals of Irreversible Thermodynamics for Coupled Transport

where the diffusivity is defined as

DOI: http://dx.doi.org/10.5772/intechopen.86607

Within this calculation, we used

where y<sup>2</sup> � � is the mean value of y2, calculated as

equation:

and

3.3.3 Langevin's picture

[7, 8]:

87

∂C <sup>∂</sup><sup>x</sup> <sup>y</sup> <sup>þ</sup>

> ∂2 C

and substitute Eq. (31) with Eq. (29). We finally derive the so-called diffusion

DB <sup>¼</sup> <sup>y</sup><sup>2</sup> � �

ðþ<sup>∞</sup> �∞ y2

> ∂C ∂t

∂C <sup>∂</sup><sup>t</sup> <sup>¼</sup> DB

<sup>y</sup><sup>2</sup> � � <sup>¼</sup>

C xð Þ¼ ; t þ δt C xð Þþ ; t

ðþ<sup>∞</sup> �∞

because yΦ is an odd function. Mathematically, Einstein's picture uses shortranged transition probability function, which does not need to be specifically known, and Taylor's expansion for a small time interval and short displacement. Conditions required for Eq. (32) are as follows: (i) transition distance is longer than the size of molecule, dx≥ O að Þ, and (ii) time interval δt is long enough to measure dx after a tremendous number of collisions with solvent molecules, satisfying δt ≫ τp, where τ<sup>p</sup> is the particle relaxation time (see Langevin's picture).

Let us consider a particle of mass m, located at x tð Þ with velocity v � dx=dt at time t. For simplicity, we shall treat the problem of diffusion in one dimension. It would be hopeless to deterministically trace all the collisions of this particle with a number of solvent molecules in series. However, these collisions can be regarded as a net force A tð Þ effective in determining the time dependence of the molecule's position x tð Þ. Newton's second law of motion can be written in the following form

which is called Langevin's equation. In Eq. (37), A tð Þ is assumed to be randomly

h iy ¼

m dv dt

and rapidly fluctuating. We multiply x on both sides of Eq. (37) to give

1 2! ∂2 C

<sup>∂</sup>x<sup>2</sup> <sup>y</sup><sup>2</sup> <sup>þ</sup> <sup>⋯</sup> (31)

<sup>∂</sup>x<sup>2</sup> (32)

<sup>2</sup>!δ<sup>t</sup> (33)

Φð Þy dy (34)

yΦð Þy dy ¼ 0 (36)

¼ �βv þ A tð Þ (37)

δt þ ⋯ (35)

and in a concise form

$$
\left< \mathbf{x}\_n^2 \right> = \left< \sum\_{i \neq j} \Delta \mathbf{x}\_i \cdot \Delta \mathbf{x}\_j \right> + \left< \sum\_{k=1}^n \Delta \mathbf{x}\_k^2 \right> = \mathbf{0} + n \Delta \mathbf{x}^2 = nl^2 \tag{25}
$$

because <sup>∑</sup><sup>i</sup>6¼<sup>j</sup> Δxi � Δxj D E <sup>¼</sup> 0 and <sup>Δ</sup>x<sup>2</sup> k � � <sup>¼</sup> ð Þ <sup>Δ</sup><sup>x</sup> <sup>2</sup> <sup>¼</sup> <sup>l</sup> 2 . In the calculation of offdiagonal terms, Δxi � Δxj can have four possible values with equal chance of ð Þ þ; þ , ð Þ þ; � , ð Þ �; þ , and ð Þ �; � . The products of the two elements in the parenthesis are þ, �, �, and þ with equal probability of 25%. Therefore, a sum of them is zero. Because n is the number of time steps, it can be replaced by t=Δt where t is the total elapsed time. The diffusion coefficient in one-dimensional space was derived in the previous section as D ¼ l 2 =2Δt. Then, the mean of squared distance at time t is calculated as

$$
\left< \mathfrak{x}^2(t) \right> = 2Dt \tag{26}
$$

and the root-mean-square distance is

$$\left| \mathfrak{x}\_{\text{rms}} = \sqrt{\langle \mathfrak{x}^2(t) \rangle} = \sqrt{2Dt} \tag{27}$$

Note that xrms is proportional to t <sup>1</sup>=<sup>2</sup> in the random walk, as compared to the constant velocity case x ¼ vt∝t 1. Then, the diffusivity for 1D is explicitly

$$D = \frac{\text{x}\_{\text{rms}}^2}{2t} = \frac{n l^2}{2n\Delta t} = \frac{l^2}{2\Delta t} \tag{28}$$

### 3.3.2 Einstein's picture

The concentration C xð Þ ; t after an infinitesimal time duration δt from t within a range dx between x and x þ dx is calculated as [6]

$$\mathbf{C}(\mathbf{x}, t + \delta t)\mathbf{d}\mathbf{x} = \mathbf{d}\mathbf{x} \int\_{-\infty}^{+\infty} \mathbf{C}(\mathbf{x} - \mathbf{y}, t)\Phi(\mathbf{y})\mathbf{d}\mathbf{y} \tag{29}$$

where Φ is the transition probability for a linear displacement y and the righthand side indicates the amount of adjacent solutes that move into the small region dx. The probability distribution satisfies

$$\int\_{-\infty}^{+\infty} \Phi(y) dy = 1\tag{30}$$

and we assume that Φ is a short ranged, even function, meaning that it is nonzero for small ∣y∣ and symmetric, Φð Þ¼ �y Φð Þy . In this case, we approximate the integrand of Eq. (29) as

Fundamentals of Irreversible Thermodynamics for Coupled Transport DOI: http://dx.doi.org/10.5772/intechopen.86607

$$\mathbf{C}(\mathbf{x} - \mathbf{y}, t) = \mathbf{C}(\mathbf{x}, t) - \frac{\partial \mathbf{C}}{\partial \mathbf{x}} \mathbf{y} + \frac{\mathbf{1}}{2!} \frac{\partial^2 \mathbf{C}}{\partial \mathbf{x}^2} \mathbf{y}^2 + \dotsb \tag{31}$$

and substitute Eq. (31) with Eq. (29). We finally derive the so-called diffusion equation:

$$\frac{\partial \mathbf{C}}{\partial t} = D\_B \frac{\partial^2 \mathbf{C}}{\partial \mathbf{x}^2} \tag{32}$$

where the diffusivity is defined as

$$D\_{\mathcal{B}} = \frac{\langle \mathcal{y}^2 \rangle}{2! \delta t} \tag{33}$$

where y<sup>2</sup> � � is the mean value of y2, calculated as

$$
\langle \chi^2 \rangle = \int\_{-\infty}^{+\infty} \chi^2 \Phi(\chi) d\chi \tag{34}
$$

Within this calculation, we used

$$\mathbf{C}(\mathbf{x}, t + \delta t) = \mathbf{C}(\mathbf{x}, t) + \frac{\partial \mathbf{C}}{\partial t} \delta t + \dotsb \tag{35}$$

and

x2 n

and in a concise form

Non-Equilibrium Particle Dynamics

because <sup>∑</sup><sup>i</sup>6¼<sup>j</sup>

previous section as D ¼ l

calculated as

x2 n � � <sup>¼</sup> <sup>∑</sup>

<sup>¼</sup> 〈Δx<sup>2</sup>

þ ⋯

Δxi � Δxj \* +

<sup>¼</sup> 0 and <sup>Δ</sup>x<sup>2</sup>

x2

<sup>x</sup>rms <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>D</sup> <sup>¼</sup> <sup>x</sup><sup>2</sup> rms <sup>2</sup><sup>t</sup> <sup>¼</sup> nl<sup>2</sup> 2nΔt

<sup>x</sup><sup>2</sup> h i ð Þ<sup>t</sup> <sup>p</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffi

The concentration C xð Þ ; t after an infinitesimal time duration δt from t within a

ðþ<sup>∞</sup> �∞

where Φ is the transition probability for a linear displacement y and the righthand side indicates the amount of adjacent solutes that move into the small region

and we assume that Φ is a short ranged, even function, meaning that it is nonzero for small ∣y∣ and symmetric, Φð Þ¼ �y Φð Þy . In this case, we approximate the

i6¼j

2

Δxi � Δxj D E

and the root-mean-square distance is

Note that xrms is proportional to t

dx. The probability distribution satisfies

range dx between x and x þ dx is calculated as [6]

C xð Þ ; t þ δt dx ¼ dx

ðþ<sup>∞</sup> �∞

constant velocity case x ¼ vt∝t

3.3.2 Einstein's picture

integrand of Eq. (29) as

86

� � <sup>¼</sup> h i <sup>ð</sup>Δx<sup>1</sup> <sup>þ</sup> <sup>⋯</sup>ΔxnÞ � ð Þ <sup>Δ</sup>x<sup>1</sup> <sup>þ</sup> <sup>⋯</sup>Δxn

<sup>þ</sup> <sup>Δ</sup>x<sup>2</sup> � <sup>Δ</sup>x<sup>1</sup> <sup>þ</sup> <sup>Δ</sup>x<sup>2</sup>

<sup>1</sup> þ Δx<sup>1</sup> � Δx<sup>2</sup> þ ⋯ þ Δx<sup>1</sup> � Δxn

<sup>þ</sup> <sup>Δ</sup>xn � <sup>Δ</sup>x<sup>1</sup> <sup>þ</sup> <sup>Δ</sup>xn � <sup>Δ</sup>x<sup>2</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> <sup>Δ</sup>x<sup>2</sup>

þ ∑ n k¼1 Δx<sup>2</sup> k � �

k � � <sup>¼</sup> ð Þ <sup>Δ</sup><sup>x</sup>

diagonal terms, Δxi � Δxj can have four possible values with equal chance of ð Þ þ; þ , ð Þ þ; � , ð Þ �; þ , and ð Þ �; � . The products of the two elements in the parenthesis are þ, �, �, and þ with equal probability of 25%. Therefore, a sum of them is zero. Because n is the number of time steps, it can be replaced by t=Δt where t is the total elapsed time. The diffusion coefficient in one-dimensional space was derived in the

<sup>2</sup> þ ⋯ þ Δx<sup>2</sup> � Δxn

<sup>2</sup> <sup>¼</sup> <sup>l</sup> 2

=2Δt. Then, the mean of squared distance at time t is

1. Then, the diffusivity for 1D is explicitly

¼ l 2

ð Þ<sup>t</sup> � � <sup>¼</sup> <sup>2</sup>Dt (26)

<sup>1</sup>=<sup>2</sup> in the random walk, as compared to the

<sup>2</sup>Dt <sup>p</sup> (27)

<sup>2</sup>Δ<sup>t</sup> (28)

C xð Þ � y; t Φð Þy dy (29)

Φð Þy dy ¼ 1 (30)

n〉

<sup>¼</sup> <sup>0</sup> <sup>þ</sup> <sup>n</sup>Δx<sup>2</sup> <sup>¼</sup> nl<sup>2</sup> (25)

. In the calculation of off-

(24)

$$\langle \boldsymbol{\chi} \rangle = \int\_{-\infty}^{+\infty} \boldsymbol{\chi} \Phi(\boldsymbol{\chi}) d\boldsymbol{\mathfrak{y}} = \mathbf{0} \tag{36}$$

because yΦ is an odd function. Mathematically, Einstein's picture uses shortranged transition probability function, which does not need to be specifically known, and Taylor's expansion for a small time interval and short displacement. Conditions required for Eq. (32) are as follows: (i) transition distance is longer than the size of molecule, dx≥ O að Þ, and (ii) time interval δt is long enough to measure dx after a tremendous number of collisions with solvent molecules, satisfying δt ≫ τp, where τ<sup>p</sup> is the particle relaxation time (see Langevin's picture).

## 3.3.3 Langevin's picture

Let us consider a particle of mass m, located at x tð Þ with velocity v � dx=dt at time t. For simplicity, we shall treat the problem of diffusion in one dimension. It would be hopeless to deterministically trace all the collisions of this particle with a number of solvent molecules in series. However, these collisions can be regarded as a net force A tð Þ effective in determining the time dependence of the molecule's position x tð Þ. Newton's second law of motion can be written in the following form [7, 8]:

$$m\frac{\mathbf{d}v}{\mathbf{d}t} = -\beta v + A(t) \tag{37}$$

which is called Langevin's equation. In Eq. (37), A tð Þ is assumed to be randomly and rapidly fluctuating. We multiply x on both sides of Eq. (37) to give

Non-Equilibrium Particle Dynamics

$$
\rho \text{xx\frac{dv}{dt}} = -\beta \text{x}v + \text{x}A(t) \tag{38}
$$

<sup>x</sup><sup>2</sup> � � <sup>¼</sup> <sup>2</sup>kBT β

Fundamentals of Irreversible Thermodynamics for Coupled Transport

DOI: http://dx.doi.org/10.5772/intechopen.86607

<sup>¼</sup> <sup>2</sup>kBTτ<sup>p</sup> β

<sup>x</sup><sup>2</sup> � � <sup>¼</sup> <sup>2</sup>kBT

<sup>x</sup>rms <sup>¼</sup> ffiffiffiffiffiffiffiffi

<sup>x</sup>rmsð Þ¼ <sup>Δ</sup><sup>t</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

If t ≫ τp, then t=τ<sup>p</sup> in the rectangular parenthesis is dominant:

cient of Brownian motion or Stokes-Einstein diffusivity is

identical to Eq. (16). The root-mean-square distance is

particle drifts for an interval Δt, where Δt ≫ τp, and then

h i f tð Þ <sup>¼</sup> <sup>1</sup>

Relationships between parameters are

Tp

fi ð Þt fj ð Þt D E <sup>¼</sup> <sup>δ</sup>ij<sup>δ</sup> <sup>t</sup> � <sup>t</sup>

ðTp 0

which is proportional to ffiffi

constant velocity motion.

t p .

proportional to ffiffi

3.3.4 Gardiner's picture

where f satisfies

and

89

ðt 0

1 � e �t 0 =τ<sup>p</sup> � �d<sup>t</sup>

t τp þ e

Stokes' law of Eq. (17) indicates β ¼ 6πηa, and, therefore, the diffusion coeffi-

DB <sup>¼</sup> kBT 6πηa

<sup>x</sup><sup>2</sup> h i <sup>p</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffi

<sup>x</sup><sup>2</sup> h i ð Þ <sup>t</sup> <sup>þ</sup> <sup>Δ</sup><sup>t</sup> � <sup>x</sup><sup>2</sup> h i ð Þ<sup>t</sup> <sup>p</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffi

Then, the time step Δt is of a macroscopic scale in that one can appreciate the movement of the particle of an order of particle radius. For a short time t ≪ τp, the mean-square distance of Eq. (48) is approximated as xrms ¼ vrmst, indicating a

Einstein's and Langevin's pictures provide identical results for xrms and DB as related to Stokes' law. On one hand, if a particle is translating with a constant velocity, its distance from the initial location is linearly proportional to the elapsed time; on the other hand, if particle is diffusing, its root-mean-square distance is

In Langevin's Eq. (37), the randomly fluctuating force can be written as

<sup>t</sup> <sup>p</sup> . Note that h i <sup>x</sup> <sup>¼</sup> 0. From an arbitrary time <sup>t</sup>, the

A tðÞ¼ αf tð Þ (53)

f tð Þdt ¼ 0 for Tp ≫ τ<sup>p</sup> (54)

<sup>0</sup> ð Þ (55)

0

� � (48)

<sup>β</sup> <sup>t</sup> � <sup>2</sup>DBt (49)

<sup>2</sup>DBt <sup>p</sup> (51)

<sup>2</sup>DBΔ<sup>t</sup> <sup>p</sup> (52)

(50)

�t=τ<sup>p</sup> � <sup>1</sup>

and take a time average of both sides during an interval τ, defined as

$$\langle \cdots \rangle = \frac{1}{\pi} \int\_{t}^{t+\tau} (\cdots) \mathbf{d}t \tag{39}$$

Then, we have after a much longer time than the particle relaxation time τ:

$$
\langle m \Big\langle \mathbf{x} \frac{\mathbf{d}v}{\mathbf{d}t} \rangle = -\beta \langle \mathbf{x}v \rangle + \langle \mathbf{x}A(t) \rangle \tag{40}
$$

Because the random fluctuating force A tð Þ is independent of the particle position x tð Þ, we calculate

$$
\langle \mathbf{x}A \rangle = \langle \mathbf{x} \rangle \langle A \rangle = \langle \mathbf{x} \rangle \cdot \mathbf{0} = \mathbf{0} \tag{41}
$$

For further derivation, we use the following identities:

$$\frac{d\mathbf{x}^2}{dt} = 2\mathbf{x}\dot{\mathbf{x}} = 2\mathbf{x}v \tag{42}$$

$$\frac{\text{d}^2 \text{x}^2}{\text{d}t^2} = \frac{\text{d}}{\text{d}t}(2\text{x}\dot{\text{x}}) = 2\text{v}^2 + 2\text{x}\text{v} \tag{43}$$

to provide

$$m\left\langle \frac{1}{2}\frac{\mathrm{d}^2\mathrm{x}^2}{\mathrm{d}t^2} - v^2 \right\rangle = -\beta \left\langle \frac{1}{2}\frac{\mathrm{d}\mathrm{x}^2}{\mathrm{d}t} \right\rangle \tag{44}$$

We let <sup>z</sup> <sup>¼</sup> <sup>d</sup>x<sup>2</sup>=d<sup>t</sup> � � and rewrite Eq. (44):

$$m\frac{d\mathbf{z}}{dt} = -\beta \left(\mathbf{z} - \frac{2k\_B T}{\beta}\right) \tag{45}$$

because the kinetic energy of this particle is equal to the thermal energy:

$$\frac{1}{2}mv^2 = \frac{1}{2}k\_BT\tag{46}$$

where kB is the Boltzmann constant. Note that the origin of the particle motion exists as the number of its collisions with solvent molecules at temperature T: If we take an initial condition of z ¼ 0 indicating either position or velocity is initially zero, then we obtain

$$z(t) = \frac{2k\_B T}{\beta} \left( \mathbf{1} - e^{-t/\tau\_p} \right) = \frac{\mathbf{d} \langle \mathbf{x}^2 \rangle}{\mathbf{d}t} \tag{47}$$

where τ<sup>p</sup> ¼ m=β is the particle relaxation time. One more integration with respect to time yields

Fundamentals of Irreversible Thermodynamics for Coupled Transport DOI: http://dx.doi.org/10.5772/intechopen.86607

$$\begin{split} \langle \mathbf{x}^2 \rangle &= \frac{2k\_B T}{\beta} \int\_0^t \left( \mathbf{1} - e^{-t'/\tau\_p} \right) \mathbf{d}t' \\ &= \frac{2k\_B T \tau\_p}{\beta} \left[ \frac{t}{\tau\_p} + e^{-t/\tau\_p} - \mathbf{1} \right] \end{split} \tag{48}$$

If t ≫ τp, then t=τ<sup>p</sup> in the rectangular parenthesis is dominant:

$$
\langle \mathbf{x}^2 \rangle = \frac{2k\_B T}{\beta} \mathbf{t} \equiv \mathbf{2} D\_B \mathbf{t} \tag{49}
$$

Stokes' law of Eq. (17) indicates β ¼ 6πηa, and, therefore, the diffusion coefficient of Brownian motion or Stokes-Einstein diffusivity is

$$D\_B = \frac{k\_B T}{6\pi\eta a} \tag{50}$$

identical to Eq. (16). The root-mean-square distance is

$$
\langle \mathfrak{x}\_{\text{rms}} = \sqrt{\langle \mathfrak{x}^2 \rangle} = \sqrt{2D\_B t} \tag{51}
$$

which is proportional to ffiffi <sup>t</sup> <sup>p</sup> . Note that h i <sup>x</sup> <sup>¼</sup> 0. From an arbitrary time <sup>t</sup>, the particle drifts for an interval Δt, where Δt ≫ τp, and then

$$
\langle \mathbf{x}\_{\rm rms}(\Delta t) = \sqrt{\langle \mathbf{x}^2(t + \Delta t) \rangle - \langle \mathbf{x}^2(t) \rangle} = \sqrt{2D\_B \Delta t} \tag{52}
$$

Then, the time step Δt is of a macroscopic scale in that one can appreciate the movement of the particle of an order of particle radius. For a short time t ≪ τp, the mean-square distance of Eq. (48) is approximated as xrms ¼ vrmst, indicating a constant velocity motion.

Einstein's and Langevin's pictures provide identical results for xrms and DB as related to Stokes' law. On one hand, if a particle is translating with a constant velocity, its distance from the initial location is linearly proportional to the elapsed time; on the other hand, if particle is diffusing, its root-mean-square distance is proportional to ffiffi t p .

### 3.3.4 Gardiner's picture

In Langevin's Eq. (37), the randomly fluctuating force can be written as

$$A(t) = q\mathbf{f}(t)\tag{53}$$

where f satisfies

$$
\langle f(t) \rangle = \frac{1}{T\_p} \int\_0^{T\_p} f(t) \mathbf{d}t = \mathbf{0} \quad \text{for} \quad T\_p \gg \tau\_p \tag{54}
$$

and

mx dv dt

m x dv dt � �

For further derivation, we use the following identities:

d2 x2 <sup>d</sup>t<sup>2</sup> <sup>¼</sup> <sup>d</sup> dt

m 1 2 d2 x2 <sup>d</sup>t<sup>2</sup> � <sup>v</sup><sup>2</sup> \* +

> m dz dt

z tðÞ¼ <sup>2</sup>kBT β

We let <sup>z</sup> <sup>¼</sup> <sup>d</sup>x<sup>2</sup>=d<sup>t</sup> � � and rewrite Eq. (44):

dx<sup>2</sup>

x tð Þ, we calculate

Non-Equilibrium Particle Dynamics

to provide

zero, then we obtain

respect to time yields

88

and take a time average of both sides during an interval τ, defined as

h i <sup>⋯</sup> <sup>¼</sup> <sup>1</sup> τ ðtþ<sup>τ</sup> t

Then, we have after a much longer time than the particle relaxation time τ:

Because the random fluctuating force A tð Þ is independent of the particle position

¼ �β

¼ �<sup>β</sup> <sup>z</sup> � <sup>2</sup>kBT

because the kinetic energy of this particle is equal to the thermal energy:

mv<sup>2</sup> <sup>¼</sup> <sup>1</sup> 2

where kB is the Boltzmann constant. Note that the origin of the particle motion exists as the number of its collisions with solvent molecules at temperature T: If we take an initial condition of z ¼ 0 indicating either position or velocity is initially

> 1 � e �t=τ<sup>p</sup> � �

where τ<sup>p</sup> ¼ m=β is the particle relaxation time. One more integration with

1 2

1 2 dx<sup>2</sup> dt � �

β � �

<sup>¼</sup> <sup>d</sup> <sup>x</sup><sup>2</sup> � �

¼ �βxv þ xA tð Þ (38)

¼ �βh i xv þ h i xA tð Þ (40)

h i xA ¼ h i x h i A ¼ h i x � 0 ¼ 0 (41)

<sup>d</sup><sup>t</sup> <sup>¼</sup> <sup>2</sup>xx\_ <sup>¼</sup> <sup>2</sup>xv (42)

ð Þ¼ <sup>2</sup>xx\_ <sup>2</sup>v<sup>2</sup> <sup>þ</sup> <sup>2</sup>xv (43)

kBT (46)

<sup>d</sup><sup>t</sup> (47)

(44)

(45)

ð Þ ⋯ dt (39)

$$
\left\langle f\_i(t)f\_j(t) \right\rangle = \delta\_{\vec{\eta}} \delta(t - t') \tag{55}
$$

Relationships between parameters are

$$
\beta = \mathsf{G}\pi\mu\mathsf{a}\_p\tag{56}
$$

C0

DOI: http://dx.doi.org/10.5772/intechopen.86607

Fundamentals of Irreversible Thermodynamics for Coupled Transport

C″ ð Þ dx

Substitution of Eqs. (70) and (71) with Eq. (67) gives

Eq. (67) is

and therefore

equation:

force F:

91

4. Dissipation rates

4.1 Energy consumption per time

particle dr under the influence of force field F is

<sup>d</sup><sup>x</sup> <sup>¼</sup> <sup>C</sup><sup>0</sup> <sup>v</sup>d<sup>t</sup> <sup>þ</sup> ffiffiffiffiffiffiffiffi

<sup>2</sup> <sup>¼</sup> <sup>C</sup>″ <sup>v</sup>d<sup>t</sup> <sup>þ</sup> ffiffiffiffiffiffiffiffi

dC xð Þ¼ C<sup>0</sup>

∂C <sup>∂</sup><sup>t</sup> <sup>¼</sup> DB

∂C <sup>∂</sup><sup>t</sup> <sup>¼</sup> DB

which can be directly obtained by replacing Eq. (65) by

using Eq. (60) of the time average, which implies that the diffusion time scale already satisfies the restricted condition of dt ≫ τp. The second term of

after dropping the second order term of dt and the first order term of dW.

2DB p dW � �≈C<sup>0</sup>

2DB p dW � �<sup>2</sup>

vdt þ C″

∂C ∂x

∂2 C <sup>∂</sup>x<sup>2</sup> <sup>þ</sup> <sup>v</sup>

which looks similar to the conventional convective diffusion equation with the sign of v reversed. Eq. (73) indicates that a group of identical particles of mass m undergoes convective and diffusive transport in the Eulerian space. A particle in the group is located at the position x at time t, moving with velocity v. This specific particle observes the concentration C of other particles nearby its position x. Therefore, Eq. (73) exists as the convective diffusion equation in the Lagrangian picture. If the particle moves with velocity v in a stationary fluid, then the motion is equivalent to particles that perform only diffusive motion within a fluid moving with �v. To emphasize the fluid velocity, we replace v with �u; then the Lagrangian convective diffusion Eq. (73) becomes the original (Eulerian) convection-diffusion

> ∂2 C <sup>∂</sup>x<sup>2</sup> � <sup>u</sup>

<sup>d</sup><sup>x</sup> ¼ �ud<sup>t</sup> <sup>þ</sup> ffiffiffiffiffiffiffiffi

In classical mechanics, work done due to an infinitesimal displacement of a

The time differentiation of Eq. (76) provides an energy consumption rate (i.e., power represented by P) as a dot product of the particle velocity v and the applied

∂C ∂x

2DB

p dW (75)

dW ¼ F � dr (76)

≈C″

vdt (70)

2DBdt (71)

(73)

(74)

DBdt (72)

$$a = \sqrt{2\beta k\_B T} = \sqrt{2D\_B^{-1}} k\_B T \tag{57}$$

$$D\_B = \frac{k\_B T}{\beta} \tag{58}$$

(See the next section for the Brownian diffusivity DB.) As such, we assume that

$$f(t)\mathbf{d}t = \mathbf{d}W(t)\tag{59}$$

where dW is the Ito-Wiener process [9, 10], satisfying

$$
\langle \mathbf{d}W \rangle = \mathbf{0} \tag{60}
$$

$$\left<\left(\mathbf{d}\mathcal{W}\right)\right>^{2} = \mathbf{d}t\tag{61}$$

Then, we can obtain the stochastic differential equation (SDE) as

$$
\rho \mathbf{u} \mathbf{d}v = [F(\mathbf{x}) - \beta v] \mathbf{d}t + a \mathbf{d}W \tag{62}
$$

The relationship between x, v, and t can be obtained as follows [11]:

$$\mathbf{dx} = \nu \mathbf{d}t \tag{63}$$

$$\mathbf{d}v = \left[\frac{F(\mathbf{x})}{m} - \frac{v}{\tau\_p}\right]\mathbf{d}t + \frac{a}{m}\mathbf{d}\mathcal{W} \tag{64}$$

Note that Eq. (63) is free from the fundamental restriction of Langevin's equation (i.e., τ<sup>p</sup> ≪ dt) by introducing the Ito-Wiener process in Eq. (64). The time interval dt can be arbitrarily chosen to improve calculation speed and/or numerical accuracy.

Eq. (63) uses the basic definition of velocity as a time derivative of the position in the classical mechanics, and Eq. (64) represents the randomly fluctuating force using the Ito-Weiner process, dW. If we keep Langevin's picture, then these two equations should have forms of

$$\mathbf{d}\mathbf{x} = \nu \mathbf{d}t + \sqrt{2D\_B} \,\mathrm{d}W \tag{65}$$

$$\mathbf{d}v = \left[\frac{F(\mathbf{x})}{m} - \frac{v}{\tau\_p}\right] \mathbf{d}t \tag{66}$$

where the random fluctuation disappears in the force balance and appears as a drift displacement, ffiffiffiffiffiffiffiffi 2DB <sup>p</sup> <sup>d</sup>W. Let C xð Þ be the concentration of particles near the position x of a specific particle. Note that x is not a fixed point in Eulerian space but a moving coordinate of a particle being tracked. An infinitesimal change of C is

$$\mathbf{d}\mathbf{C}(\mathbf{x}) = \mathbf{C}^{\prime}\mathbf{d}\mathbf{x} + \frac{\mathbf{1}}{2!}\mathbf{C}^{\prime}\mathbf{d}\mathbf{x}^{2} + \cdots \tag{67}$$

where

$$\mathbf{C}' = \frac{\partial \mathbf{C}}{\partial \mathbf{x}}\tag{68}$$

$$\mathbf{C}^{'} = \frac{\partial^2 \mathbf{C}}{\partial \mathbf{x}^2} \tag{69}$$

The first term of Eq. (67) is

Fundamentals of Irreversible Thermodynamics for Coupled Transport DOI: http://dx.doi.org/10.5772/intechopen.86607

$$\mathbf{C}'\mathbf{dx} = \mathbf{C}'\left(\nu\mathbf{d}t + \sqrt{2D\_B}\,\mathrm{d}\mathcal{W}\right) \approx \mathbf{C}'\nu\mathrm{d}t\tag{70}$$

using Eq. (60) of the time average, which implies that the diffusion time scale already satisfies the restricted condition of dt ≫ τp. The second term of Eq. (67) is

$$\mathbf{C}^{\prime}(\mathbf{dx})^{2} = \mathbf{C}^{\prime}\left(\nu\mathbf{d}t + \sqrt{2D\_{B}}\mathbf{d}W\right)^{2} \approx \mathbf{C}^{\prime}2D\_{B}\mathbf{d}t \tag{71}$$

after dropping the second order term of dt and the first order term of dW. Substitution of Eqs. (70) and (71) with Eq. (67) gives

$$\mathbf{d}C(\mathbf{x}) = \mathbf{C}^{\prime}\boldsymbol{\nu}\mathbf{d}\mathbf{t} + \mathbf{C}^{\prime}D\_{B}\mathbf{d}\mathbf{t} \tag{72}$$

and therefore

β ¼ 6πμap (56)

f tð Þdt ¼ dW tð Þ (59)

h i dW ¼ 0 (60) h i ð Þ <sup>d</sup><sup>W</sup> <sup>2</sup> <sup>¼</sup> <sup>d</sup><sup>t</sup> (61)

dx ¼ vdt (63)

p dW (65)

dt (66)

<sup>d</sup>x<sup>2</sup> <sup>þ</sup> <sup>⋯</sup> (67)

<sup>∂</sup>x<sup>2</sup> (69)

(68)

dW (64)

mdv ¼ ½ � F xð Þ� βv dt þ αdW (62)

kBT (57)

<sup>β</sup> (58)

ffiffiffiffiffiffiffiffiffiffi 2D�<sup>1</sup> B

q

DB <sup>¼</sup> kBT

(See the next section for the Brownian diffusivity DB.) As such, we assume that

<sup>α</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>2</sup>βkBT <sup>p</sup> <sup>¼</sup>

where dW is the Ito-Wiener process [9, 10], satisfying

Then, we can obtain the stochastic differential equation (SDE) as

The relationship between x, v, and t can be obtained as follows [11]:

<sup>m</sup> � <sup>v</sup> τp

<sup>d</sup><sup>x</sup> <sup>¼</sup> <sup>v</sup>d<sup>t</sup> <sup>þ</sup> ffiffiffiffiffiffiffiffi

<sup>d</sup><sup>v</sup> <sup>¼</sup> F xð Þ

<sup>d</sup>C xð Þ¼ <sup>C</sup><sup>0</sup>

2DB

<sup>p</sup> <sup>d</sup>W. Let C xð Þ be the concentration of particles near the

<sup>m</sup> � <sup>v</sup> τp

� �

where the random fluctuation disappears in the force balance and appears as a

position x of a specific particle. Note that x is not a fixed point in Eulerian space but a moving coordinate of a particle being tracked. An infinitesimal change of C is

> dx þ 1 2! C″

<sup>C</sup><sup>0</sup> <sup>¼</sup> <sup>∂</sup><sup>C</sup> ∂x

<sup>C</sup>″ <sup>¼</sup> <sup>∂</sup><sup>2</sup>

C

Note that Eq. (63) is free from the fundamental restriction of Langevin's equation (i.e., τ<sup>p</sup> ≪ dt) by introducing the Ito-Wiener process in Eq. (64). The time interval dt can be arbitrarily chosen to improve calculation speed and/or numerical accuracy. Eq. (63) uses the basic definition of velocity as a time derivative of the position in the classical mechanics, and Eq. (64) represents the randomly fluctuating force using the Ito-Weiner process, dW. If we keep Langevin's picture, then these two

<sup>d</sup><sup>t</sup> <sup>þ</sup> <sup>α</sup> m

� �

<sup>d</sup><sup>v</sup> <sup>¼</sup> F xð Þ

equations should have forms of

Non-Equilibrium Particle Dynamics

drift displacement, ffiffiffiffiffiffiffiffi

The first term of Eq. (67) is

where

90

2DB

$$\frac{\partial \mathbf{C}}{\partial t} = D\_B \frac{\partial^2 \mathbf{C}}{\partial \mathbf{x}^2} + v \frac{\partial \mathbf{C}}{\partial \mathbf{x}} \tag{73}$$

which looks similar to the conventional convective diffusion equation with the sign of v reversed. Eq. (73) indicates that a group of identical particles of mass m undergoes convective and diffusive transport in the Eulerian space. A particle in the group is located at the position x at time t, moving with velocity v. This specific particle observes the concentration C of other particles nearby its position x. Therefore, Eq. (73) exists as the convective diffusion equation in the Lagrangian picture. If the particle moves with velocity v in a stationary fluid, then the motion is equivalent to particles that perform only diffusive motion within a fluid moving with �v. To emphasize the fluid velocity, we replace v with �u; then the Lagrangian convective diffusion Eq. (73) becomes the original (Eulerian) convection-diffusion equation:

$$\frac{\partial \mathbf{C}}{\partial t} = D\_B \frac{\partial^2 \mathbf{C}}{\partial \mathbf{x}^2} - u \frac{\partial \mathbf{C}}{\partial \mathbf{x}} \tag{74}$$

which can be directly obtained by replacing Eq. (65) by

$$\mathbf{dx} = -\mathbf{u}\mathbf{d}t + \sqrt{2D\_B}\mathbf{d}W \tag{75}$$
