Relaxation phenomenon

Japanese: 緩和現象 - かんわげんしょう

When a system surrounded by the outside world is in a state of thermal equilibrium, if the outside world suddenly changes, the system also changes (responds) and moves toward a state of thermal equilibrium determined by the new external conditions. This phenomenon in which it takes time for a system to reach a new state of thermal equilibrium in response to a change (action) in the outside world is called a relaxation phenomenon. For example, if a gas occupies one side of a container with a partition, and a hole is made in the partition, the gas diffuses to the other side, which is kept in a vacuum, and eventually the gas becomes of uniform density throughout the container, reaching a state of thermal equilibrium.

When a magnetic field is suddenly applied to a magnetic material that initially has no magnetization, the magnetization of the magnetic material gradually increases, and eventually the material becomes magnetic with a constant magnetization. A similar phenomenon also occurs when an electric field is suddenly applied to a dielectric material (insulator) and electric polarization occurs in the dielectric. This is also a relaxation phenomenon to a new equilibrium state under new external conditions (magnetic field, electric field). When an electric field is applied to a conductor such as metal, current flows instantly, and it appears that the current stops immediately when the electric field is turned off, but this is also a relaxation phenomenon.

When the response to an external field is proportional to the magnitude of the field, it is called a linear response. Generally, when the external field is small, the response is linear, but when the external field is large, the response is not necessarily proportional to the magnitude of the field. In this case, it is called a nonlinear response.

[Fukuro Ono]

Dielectric relaxation

Let us consider the change in electric polarization of a dielectric due to an electric field. Let us assume that the polarization is 0 initially when the electric field E = 0, and that at time t = 0, the electric field E is suddenly applied. If the change in polarization over time is then represented as P ( t ), the polarization P ( t ), which was initially 0, gradually approaches the equilibrium value _Peq ( E ) under the electric field E. This is shown in Figure A. Conversely, if the electric field of a dielectric placed in an electric field is suddenly turned off, the polarization relaxes from the value of _Peq ( E ) to 0. This is shown in Figure B. Expressed as a formula, when an electric field is suddenly applied,
P ( t )＝ P _eq ( E )(1− e ^{- t /τ} ) (1)
When the electric field is suddenly turned off, P ( t ) = _Peq ( E ) e ^{- t /τ} (2)
Therefore, in either case, the polarization will exponentially relax toward the new thermal equilibrium value. This is called Debye relaxation, and τ (tau) is called the relaxation time. This type of relaxation is commonly seen in many physical phenomena.

When an electric field E is applied, let us assume that the time change of polarization P ( t ) obeys the following differential equation:

Here, α (alpha) is a constant. After a sufficient amount of time has passed, P (∞) becomes a constant value. In the above formula,

Then, we obtain P (∞)＝ αE . Since this value is the thermal equilibrium value of polarization under an electric field E , _Peq ( E )＝ P (∞)＝ αE . From this, we can see that α is the electric polarizability.

It is easy to verify that when the initial conditions are P (0)＝0, t ＞0 and E ≠0, the solution to equation (3) is given by equation (1). It can also be seen that if the initial conditions are P (0)＝ αE , t ＞0 and E ＝0, then equation (2) is a solution to the same equation. Therefore, if P ( t ) is a Debye-type relaxation, its time change satisfies equation (3).

[Fukuro Ono]

Debye-type relaxation and absorption of metallic current

The change in current over time when an electric field is applied to an electrical conductor (metal) also shows a similar Debye type relaxation. The flow of electrons in a metal is attenuated by collisions, so the current due to electrons also decays in the same way. If the Debye relaxation time is τ, the current j ( t ) in an electric field E is

It is thought that the time change is similar to that of polarization. Here, σ (sigma) is the electrical conductivity. The steady-state value of the current is j _eq ( E ) = σ E
This is Ohm's law.

Let us consider the response when an oscillating electric field is applied. If the external oscillating electric field is expressed as E ( t )＝ E0cos ( ωt ₎ , the steady oscillating current j ( t ) oscillates at the same frequency, but j ( t )＝ j '(ω)cos( ωt ).
+ j "(ω)sin( ωt ) (5)
As shown above, the j '(ω) component oscillates in the same phase as the external field, and the j "(ω) component oscillates 90 degrees out of phase. Substituting this equation into equation (4) and solving it, we get

The energy absorption rate per unit time is W ( t )＝ j ( t ) E ( t ), so if we average it over one period, we can see that only the component j '(ω) that oscillates in phase contributes to absorption. The value is

Figure C shows the frequency dependence of absorption intensity. This is called an absorption type or Debye type absorption curve.

[Fukuro Ono]

Lorentz type relaxation and absorption

Let's consider the case of a system with damping due to a resistive force proportional to the speed of simple harmonic motion. When steady vibration occurs under an oscillating external force, if the external force is suddenly stopped, the vibration will not simply dampen as in the Debye type, but will dampen while oscillating. This is called resonant relaxation.

Consider a damped oscillating system with a resonant frequency ω ₀ , where the damping term is given by - m γ v , where m is the mass of the object and v is the velocity. In such a system, the rate of energy absorption per unit time due to an oscillating external force f ( t ) = f ₀ cosωt is proportional to v ( t ) f ( t ), and unlike the Debye type, its time average is proportional to the response component of the displacement x that oscillates in phase with the external force, but is instead proportional to the response component shifted by 90 degrees. The form is

where f ₀ is the magnitude of the external force. Figure D shows the frequency dependence of this absorption curve due to the external field. Unlike the Debye type, it can be seen that there is an absorption peak at the resonance frequency ω _0. This is called a resonance type or Lorentz type absorption curve. In this way, the relaxation time can be determined from the width of the absorption curve for the vibration external field without having to directly calculate it from the relaxation function.

[Fukuro Ono]

Various types of palliative care

In general, when a physical quantity P has a value ΔP = P - Peq that is slightly different from the thermal equilibrium value due to an external field or fluctuation, if _it naturally relaxes (returns) to the equilibrium value,

If we follow this equation, we can see that this is Debye relaxation.

When the state approaches thermal equilibrium from a state slightly away from thermal equilibrium, the relaxation is often Debye relaxation, but it may not be possible to describe it with a single relaxation time. Also, when approaching a critical point or in a spin glass transition, the relaxation may not be of the Debye type, but may be of the power type t ^{- s} or logarithmic type log t .

[Fukuro Ono]

Relaxation function (change in polarization when an electric field is suddenly applied to a dielectric) [Figure A]
©Shogakukan ">

Relaxation function (when an electric field is suddenly applied to a dielectric material...

Relaxation function (change in polarization when the electric field is suddenly removed) (Figure B)
©Shogakukan ">

Relaxation function (when the electric field is suddenly removed)

Frequency dependence of the absorption curve of Debye relaxation (Fig. C)
The absorption peaks at ω = 0 and decreases slowly with ω .

Frequency dependence of the absorption curve for Debye relaxation […

Frequency dependence of the absorption curve of Lorentzian relaxation (Fig. D)
The absorption peak is at the resonance frequency ω of the simple harmonic motion, and the absorption width becomes wider with an increase in the resistance coefficient γ .

Frequency dependence of the absorption curve for Lorentzian relaxation…

Source: Shogakukan Encyclopedia Nipponica About Encyclopedia Nipponica Information | Legend

Japanese:

外界に囲まれている体系が熱平衡状態にあるとき、外界が急に変化すると、体系も変化（応答）し、新たな外的条件で決まる熱平衡状態に向かっていく。このような、外界の変化（作用）に対して、体系が新しい熱平衡状態になるのに時間がかかる現象を緩和現象という。たとえば、仕切り壁のある容器内の一方を占めていた気体が、仕切り壁に穴を開けると、真空に保たれた他方へ拡散し、やがて容器全体で一様な密度をもった気体になり、熱平衡状態になる。

　初め磁化をもたない磁性体に急に磁場をかけると、磁性体の磁化は徐々に増加し、やがて一定の磁化をもった磁性体になる。また誘電体（絶縁体）に急に電場をかけると、誘電体に電気分極がおきるときも同様な現象が生じる。これも新しい外的条件（磁場、電場）での新しい平衡状態への緩和現象である。金属のような導体に電場をかけると、即座に電流が流れ、電場を切るとすぐ電流が止まるようにみえるが、やはり緩和のある現象である。

　外場に対する応答が外場の大きさに比例するとき、線形応答とよばれる。一般に外場が小さいときは線型応答になるが、外場が大きいとき、応答が外場の大きさに比例するとは限らない。このときは非線形応答とよばれている。

［小野昱郎］

誘電体の緩和

誘電体の電場による電気分極の変化を考えてみよう。始め電場E＝0では分極が0で、時刻t＝0で、急に電場Eが加えられたとしよう。そのとき分極の時間変化をP(t)で表すと、始め0であった分極P(t)は電場E下での平衡値P_eq(E)にしだいに近づく。これを図Aに示す。また逆に電場中に置かれた誘電体の電場を急に切ると、分極はP_eq(E)の値から0に緩和する。これを図Bに示した。これを式で表すと、電場を急にかけたとき、
　　P(t)＝P_eq(E)(1－e^-t/τ)　　(1)
また電場を急に切ったとき
　　P(t)＝P_eq(E)e^-t/τ　　(2)
となる。したがって、いずれの場合も分極は新しい熱平衡値に向かって指数関数的に緩和することになる。これをデバイ緩和といい、τ(タウ)は緩和時間とよばれている。このような緩和の仕方は多くの物理現象に共通にみられるものである。

　電場Eをかけたとき、分極P(t)の時間変化が次の微分方程式に従うとしよう。

　ここで、α(アルファ)は定数である。十分時間がたったときは、P(∞)は一定の値になる。前記の式で

とおけば、P(∞)＝αEが得られる。この値は電場Eの下での分極の熱平衡値であるから、P_eq(E)＝P(∞)＝αEである。このことから、αは電気分極率であることがわかる。

　初期条件がP(0)＝0、t＞0でE≠0のとき、(3)の方程式の解は、(1)式で与えられることは容易に確かめられる。また、初期条件がP(0)＝αE、t＞0でE＝0とすれば、(2)式が同じ方程式の解であることもわかる。したがって、P(t)がデバイ型の緩和とすれば、その時間変化は(3)の方程式を満足することがわかる。

［小野昱郎］

金属電流のデバイ型緩和・吸収

電気の導体（金属）に電場をかけたときの電流の時間変化も同じようなデバイ型緩和を示す。金属中の電子の流れは衝突によって減衰するから、電子による電流も同じように減衰する。デバイの緩和時間をτとすれば、電場E中での電流j(t)は

のように、分極と同じような時間変化をすると考えられる。ここで、σ(シグマ)は電気伝導率である。電流の定常値は
　　j_eq(E)＝σE
と与えられる。これはオームの法則である。

　以下では振動する電場をかけたときの応答を考えてみよう。外部より与えられた振動電場をE(t)＝E₀cos(ωt)のように表すと、定常振動電流j(t)は同じ振動数で振動するが
　　j(t)＝j'(ω)cos(ωt)
　　　＋j"(ω)sin(ωt)　　(5)
のように外場と同じ位相で振動するj'(ω)の成分と90度ずれた位相で振動するj"(ω)の成分をもつ。この式を(4)式に代入して解けば、

が得られる。単位時間当りのエネルギー吸収率はW(t)＝j(t)E(t)であるから、1周期で平均をすれば、同位相で振動する成分j'(ω)のみが吸収に寄与することがわかる。その値は

である。図Cに吸収の強度の振動数依存性を示す。これを吸収型またはデバイ型吸収曲線という。

［小野昱郎］

ローレンツ型緩和・吸収

単振動で速度に比例した抵抗力による減衰がある系の場合を考えてみよう。振動する外力のもとで定常的に振動がおこっているとき、急に外力を止めると、デバイ型のように単純に減衰するのではなく、振動しながら減衰していくことがわかる。これを共鳴型緩和とよぶ。

　共鳴振動数ω₀をもつ減衰振動する系で、減衰項が－mγvで与えられるとしよう。ここで、mは物体の質量、vは速度である。このような系では、振動外力f(t)＝f₀cosωtによるエネルギーの単位時間当りの吸収率は、v(t)f(t)に比例し、その時間平均はデバイ型と異なり、外力と同じ位相で振動する変位xの応答成分ではなく、90度ずれた応答成分に比例する。その形は

で表される。ここで、f₀は外力の大きさである。この吸収曲線の外場による振動数依存性を図Dに示す。デバイ型と違って、共鳴振動数ω₀に吸収のピークがあることがわかる。これを共鳴型またはローレンツ型吸収曲線という。このように緩和時間を緩和関数から直接求めなくても、振動外場に対する吸収曲線の幅から決めることもできる。