Quantifying non-Markovianity in magnetization dynamics via entropy production rates

This paper demonstrates that while the standard stochastic Landau-Lifshitz-Gilbert equation exhibits strictly positive entropy production, inertial and open-system extensions display temporarily negative rates indicative of non-Markovianity, with the open-system model showing the highest degree of non-Markovianity across various conditions.

The Gist

Imagine you are watching a spinning top. Usually, we think of how it spins as a simple, predictable process: it wobbles, slows down due to friction, and eventually stops. In physics, this "simple" view is called Markovian. It means the top's future depends only on how it is spinning right now. It has no memory of how it spun a second ago; the past is gone.

However, in the ultra-fast world of tiny magnets (like those in your hard drive or phone), things get weird. On timescales as short as a trillionth of a second (picoseconds), these magnets don't just "forget" the past. They remember it. This is called non-Markovian behavior.

This paper is like a detective story where the authors try to figure out which mathematical rules best describe these tiny magnets and how much they remember their past. They use a clever tool called Entropy Production to solve the case.

The Detective Tool: The "Energy Bill" (Entropy)

To understand the paper, let's use an analogy of a household energy bill.

• Entropy Production is like the "cost" of running a machine. In a normal, predictable world (Markovian), the machine always consumes energy. The bill is always positive. You never get money back from the power company just for running the machine.
• Negative Entropy is like getting a refund. If the "energy bill" goes negative, it means the machine is briefly pulling energy back from its surroundings. This only happens if the machine has a "memory" of what happened before and uses that memory to cheat the system temporarily.

The authors' main discovery is: If you see a negative energy bill, you know the system has a memory (it's non-Markovian).

The Three Suspects

The paper compares three different mathematical "suspects" (equations) that scientists use to describe how magnets spin:

1. The Standard LLG (The "Simpleton"):

   • The Analogy: This is the classic spinning top. It has friction and noise, but it has no memory.
   • The Result: The authors proved mathematically and showed with computers that this model always has a positive energy bill. It never gets a refund. It is purely "forgetful" (Markovian). If you use this model, you assume the magnet has no memory.

2. The Inertial LLG (The "Heavy Top"):

   • The Analogy: Imagine a spinning top that is so heavy it has a lot of "inertia." When you push it, it doesn't just turn; it wobbles and overshoots because it's heavy. It takes a moment to react.
   • The Result: This model sometimes gets a negative energy bill (a refund), but only under specific conditions. If the magnet is aligned perfectly with the magnetic field, it acts like a simple top. But if it's tilted at an angle, the "heaviness" makes it remember the past, and it shows non-Markovian behavior.

3. The Open-System LLG (The "Smart Magnet"):

   • The Analogy: This is the most complex suspect. Imagine the spinning top is sitting in a thick, sticky fluid (like honey) that is also vibrating. The fluid doesn't just resist the top; it "talks" to it. The top pushes the fluid, and the fluid pushes back later with a delay. This delay is the "memory."
   • The Result: This model gets the biggest refunds (the most negative entropy). It is the most "rememberful" of all. The authors found that this model consistently shows the strongest signs of non-Markovian behavior, regardless of how the magnet is tilted.

The Big Reveal

The authors ran thousands of computer simulations to see which model matches reality best.

• They found that the Standard LLG is too simple. It can't explain the weird, fast oscillations seen in recent experiments with real magnets.
• The Inertial LLG is better, but it still misses some of the "memory" effects.
• The Open-System LLG is the winner. It captures the "colored noise" (the sticky, vibrating fluid) and the memory kernel (the delayed feedback).

Why does this matter?
In the past, scientists often used the simple "forgetful" model because it was easy to calculate. But this paper shows that on ultra-fast timescales (like those used in next-generation super-fast computers), the magnet does remember its past. If you ignore this memory, your predictions will be wrong.

The Takeaway

Think of the magnet not as a lonely spinning top, but as a dancer in a crowded room.

• The Standard Model thinks the dancer is alone in an empty room.
• The Open-System Model realizes the dancer is bumping into people, getting pushed back, and reacting to the crowd's movement a split second later.

The paper proves that to understand the true dance of magnets at lightning speeds, we must use the model that accounts for the crowd (the environment) and the delay (the memory). The "energy bill" (entropy) is the receipt that proves the dancer is interacting with the crowd, not just spinning in a void.

Technical Summary

1. Problem Statement

The dynamics of magnetization in ferromagnets on ultrafast (picosecond) timescales are traditionally described by the stochastic Landau-Lifshitz-Gilbert (LLG) equation. However, recent experiments have revealed complex behaviors, such as nutation (inertial oscillations) and higher-order oscillations, which the standard LLG equation cannot capture.

• The Gap: While extensions like the inertial LLG (iLLG) and open-system LLG (os-LLG) equations have been proposed to explain these phenomena, there is a lack of rigorous, quantitative frameworks to distinguish between Markovian (memoryless) and non-Markovian (memory-dependent) dynamics in these specific contexts.
• The Core Question: Can thermodynamic entropy production rates (EPR) serve as a reliable indicator to detect and quantify the degree of non-Markovianity in different magnetization dynamics models?

2. Methodology

The authors employ a combination of analytical proofs and numerical simulations to compare three distinct magnetization dynamics equations:

1. Standard LLG: The conventional stochastic equation.
2. Inertial LLG (iLLG): Includes an inertial term to account for nutation.
3. Open-System LLG (os-LLG): Derived from an open quantum system approach, featuring a memory kernel and colored noise.

Key Technical Approaches:

• Entropy Production Rate (EPR): The study utilizes the time derivative of the relative entropy (Kullback-Leibler divergence) between the system's instantaneous probability distribution $p(t)$ and the thermal equilibrium state $\pi_\beta$.
  • Theorem: Based on work by Strasberg and Esposito, if a classical undriven system satisfies local detailed balance, a negative EPR ($\dot{\Sigma}(t) < 0$) strictly indicates non-Markovian evolution.
• Analytical Bounds: The authors derive the Fokker-Planck equations for each model to prove analytically whether the dynamics are contractive (guaranteeing $\dot{\Sigma} \geq 0$) or non-contractive.
• Numerical Simulation:
  • Simulated 50,000 trajectories for Cobalt thin films using realistic parameters ($T_{sim}=0.1$, $\alpha=0.15$, $\tau=120$ fs).
  • Two initial configurations were tested: Parallel (anti-parallel alignment with field) and Canted (45° angle with a stronger field).
  • Probability distributions were mapped to spherical coordinates and discretized into bins to calculate relative entropy and EPR.
• Quantification Measures: Two specific metrics were defined to quantify non-Markovianity:
  1. Negative EPR Window ($N_A$): The integrated magnitude of negative entropy production.
  2. Relative EPR Window ($\bar{N}_A$): The negative EPR window normalized by the total absolute EPR, providing a scale-independent measure between 0 and 1.

3. Key Contributions

• Analytical Proof of Contractivity for LLG: The authors rigorously prove that the standard LLG equation always yields a strictly positive entropy production rate ($\dot{\Sigma} \geq 0$), confirming it describes purely Markovian dynamics regardless of initial conditions.
• Demonstration of Non-Markovianity in Extended Models: They analytically and numerically demonstrate that both iLLG and os-LLG equations can exhibit negative entropy production rates, signaling non-Markovian evolution due to memory effects and information backflow from the bath.
• Quantitative Hierarchy of Non-Markovianity: The study establishes a clear hierarchy:
  • LLG: Zero non-Markovianity.
  • iLLG: Non-Markovianity is conditional (observed only in the "canted" configuration where torque is high).
  • os-LLG: Exhibits the highest magnitude of non-Markovianity in all tested configurations, driven by the memory kernel.

4. Results

• Entropy Production Behavior:
  • LLG: $\dot{\Sigma}(t)$ remains strictly positive for both parallel and canted configurations.
  • iLLG: $\dot{\Sigma}(t)$ stays positive for the parallel configuration but becomes temporarily negative for the canted configuration. This suggests that inertial effects alone induce non-Markovianity only under specific high-torque conditions.
  • os-LLG: $\dot{\Sigma}(t)$ oscillates rapidly between positive and negative values for both configurations. The magnitude of the negative excursions is significantly larger than in the iLLG case.
• Quantitative Metrics:
  • For the canted configuration, the os-LLG equation showed a negative EPR window ($N_A \sim 30$) roughly one order of magnitude larger than the iLLG equation ($N_A \sim 2$).
  • The relative EPR window ($\bar{N}_A$) confirmed that os-LLG non-Markovianity is robust and less dependent on initial conditions compared to iLLG.
• Timescales: Non-Markovian effects (negative EPR) are transient, occurring primarily within the first 10–15 picoseconds (ultrafast regime). At longer times ($t > 15$ ps), all systems converge to monotonic decay, approaching the Markovian limit as the system equilibrates.

5. Significance

• Theoretical Validation: The paper provides a thermodynamic justification for using the os-LLG equation. It demonstrates that if experimental data or simulations show negative entropy production, the standard LLG framework is insufficient, and memory effects (os-LLG) must be included.
• Experimental Relevance: The findings align with recent ultrafast experiments on crystalline cobalt thin films that observed multiple oscillations. The os-LLG equation is identified as the most accurate theoretical framework for these phenomena.
• New Diagnostic Tool: The authors propose entropy production rates as a robust, experimentally accessible metric (potentially via hot-electron bolometers) to detect non-Markovian dynamics in magnetic materials, opening new avenues for studying ultrafast spintronics and quantum thermodynamics.
• Generalizability: The methodology of using negative EPR to detect non-Markovianity in rotational Brownian motion offers a general tool for analyzing complex stochastic systems beyond magnetism.

In conclusion, this work bridges the gap between stochastic thermodynamics and ultrafast magnetism, proving that the inclusion of memory kernels (os-LLG) is essential for accurately describing the non-Markovian nature of magnetization dynamics on picosecond timescales.