AMEP: The Active Matter Evaluation Package for Python
Lukas Hecht Kay-Robert Dormann Kai Luca Spanheimer Mahdieh Ebrahimi Malte Cordts Suvendu Mandal Aritra K. Mukhopadhyay Benno Liebchen


AMEP: The Active Matter Evaluation Package for Python]AMEP: The Active Matter Evaluation Package for Python


lukas.hecht@pkm.tu-darmstadt.de
 
	Institute for Condensed Matter Physics, Department of Physics, Technische Universität Darmstadt, Hochschulstr. 8, 64289 Darmstadt, Germany


 
	Institute for Condensed Matter Physics, Department of Physics, Technische Universität Darmstadt, Hochschulstr. 8, 64289 Darmstadt, Germany


 
	Institute for Condensed Matter Physics, Department of Physics, Technische Universität Darmstadt, Hochschulstr. 8, 64289 Darmstadt, Germany


 
	Institute for Condensed Matter Physics, Department of Physics, Technische Universität Darmstadt, Hochschulstr. 8, 64289 Darmstadt, Germany


 
	Institute for Condensed Matter Physics, Department of Physics, Technische Universität Darmstadt, Hochschulstr. 8, 64289 Darmstadt, Germany


 
	Institute for Condensed Matter Physics, Department of Physics, Technische Universität Darmstadt, Hochschulstr. 8, 64289 Darmstadt, Germany


 
	Institute for Condensed Matter Physics, Department of Physics, Technische Universität Darmstadt, Hochschulstr. 8, 64289 Darmstadt, Germany


benno.liebchen@pkm.tu-darmstadt.de
 
	Institute for Condensed Matter Physics, Department of Physics, Technische Universität Darmstadt, Hochschulstr. 8, 64289 Darmstadt, Germany










The Active Matter Evaluation Package () is a Python library for analyzing simulation data of particle-based and continuum simulations. It provides a powerful and simple interface for handling large data sets and for calculating and visualizing a broad variety of observables that are relevant to active matter systems. Examples range from the mean-square displacement and the structure factor to cluster-size distributions, binder cumulants, and growth exponents.  is written in pure Python and is based on powerful libraries such as NumPy, SciPy, Matplotlib, and scikit-image. Computationally expensive methods are parallelized and optimized to run efficiently on workstations, laptops, and high-performance computing architectures, and an HDF5-based data format is used in the backend to store and handle simulation data as well as analysis results.  provides the first comprehensive framework for analyzing simulation results of both particle-based and continuum simulations (as well as experimental data) of active matter systems. In particular,  also allows it to analyze simulations that combine particle-based and continuum techniques such as used to study the motion of bacteria in chemical fields or for modeling particle motion in a flow field.  is available at <https://amepproject.de> and can be installed via conda and pip.


[
    Benno Liebchen
    April 29, 2024
==================















§ INTRODUCTION


Computer simulations are a powerful method to investigate and understand the physical properties of soft matter and biological systems. In particular, molecular dynamics (MD) simulations stand out as an indispensable tool to determine the microscopic dynamics and structural properties of molecular systems comprising bio-molecules <cit.>, polymer electrolytes <cit.>, liquid crystals <cit.>, or confined liquids <cit.> for example. These systems exhibit notable relevance in both industrial and medical applications <cit.>. Expanding beyond atomistic and systematically coarse-grained MD simulations involving suitable force fields, Brownian dynamics (BD) simulations have extensively been used over the past two decades especially also for modelling active matter systems. In such systems, the individual constituents consume energy from their environment and use it to self propel and navigate through complex surroundings <cit.>. Examples of active matter systems can be observed across scales from microscopic entities such as bacteria, algae, and synthetic microswimmers <cit.> to fishes, birds, drones, and human crowds on the macroscale <cit.>. These out-of-equilibrium systems exhibit striking collective phenomena such as phase separation (where the system selects a density) <cit.>, non-equilibrium pattern formation (where the system selects a length scale) <cit.>, or other ordering transitions such as flocking showing (long-range) orientational ordering <cit.>.

To effectively model active matter systems, different computational approaches provide distinct advantages <cit.>. Particle-based models such as the active Brownian particle (ABP) model <cit.>, solved numerically using BD simulations for example, have proven valuable for investigating collective phenomena such as motility-induced phase separation (MIPS) <cit.> as well as the dynamics and local order of the involved individual particles <cit.>. Conversely, continuum models such as the active model B+ <cit.> enable us to understand collective phenomena over larger length and time scales by studying the evolution of particle densities  <cit.>. Therefore, it is a common approach to start from a particle-based model and subsequently derive a continuum model via coarse-graining techniques <cit.>. Moreover, the integration of particle-based and continuum models has emerged as a promising approach <cit.>, especially for scenarios such as modeling the interaction of active particles with a surrounding fluid or another medium that is quasi-continuous on the scale of the considered particles, as exemplified by the motion of bacteria or synthetic Janus colloids in a self-produced concentration field <cit.>. These approaches are also particularly important for modeling artificial microswimmers such as active colloids in the presence of chemical fields <cit.>, exhibiting phenomena such as chemotaxis <cit.>.

To gain physical insights from the data resulting from numerical solutions, a comprehensive array of analysis techniques is required. The mean-square displacement and time correlation functions such as the orientational autocorrelation function are popular observables to achieve insights into the dynamical aspects <cit.>, while spatial correlation functions such as the radial/pair distribution function and the structure factor are frequently considered to provide information about spatial structures <cit.>. Other observables such as entropy production and kinetic temperature are also considered frequently to analyze the collective behavior of the system under investigation <cit.>, and together with statistical analysis tools such as binder cumulants used in finite size scaling analyses <cit.> and cluster analyses including cluster growth exponents <cit.>, these observables allow us to characterize the non-equilibrium phase diagram and critical dynamics of active matter systems. Additionally, local order analyses such as local density and bond orientational order parameters offer insights into the system's local structure and symmetries <cit.>. While existing analysis packages such as freud <cit.>, MDAnalysis <cit.>, VMD <cit.>, MDTraj <cit.>, Ovito <cit.>, Pytim <cit.>, LOOS <cit.>, and MMTK <cit.> offer the possibility to calculate such observables, they are mostly inspired by and optimized for the MD simulation community and fall short in capturing all relevant observables for active matter systems in a single package. Additionally, they lack the possibility to handle continuum simulation data. Therefore, a new library is required that (i) consolidates the essential observables needed to analyze active matter simulation data, (ii) provides a unified platform for analyzing both particle-based and continuum simulation data, and (iii) seamlessly integrates data formats widely-used in computational physics through an application programming interface (API).

In this paper, we introduce the Active Matter Evaluation Package (), a unified framework for analyzing MD simulation, BD simulation, and continuum simulation data with a specific focus on soft and active matter systems.  provides an optimized framework for loading, storing, and evaluating simulation data based on particle trajectories and the time evolution of continuum fields. This framework is based on an optimized HDF5 file format <cit.> which is optimal for long-term storage purposes and for handling data of large-scale simulations <cit.>. Computationally expensive methods are parallelized and  selectively loads only the data into the main memory which is necessary for the current computational step. This ensures efficient operation on high-performance computing architectures, workstations, and laptops.  is written in pure Python and provides a user-friendly, easy-to-learn Python API that interfaces with common tools used in computational physics via NumPy arrays <cit.>. Based on common Python libraries such as NumPy <cit.>, SciPy <cit.>, Matplotlib <cit.>, and scikit-image <cit.>,  provides a powerful toolbox for calculating spatial and temporal correlation functions, visualizing and animating simulation results, and coarse-graining particle-based simulations, which makes it possible to easily analyze the dynamics and structure of both particle-based and continuum simulation data.

The paper is organized as follows: We first demonstrate a minimal example on how to use  to load, analyze, and visualize simulation data in Section <ref>. Second, we give a brief overview on the structure and design of  in Section <ref>. Finally, in Sections <ref> and <ref>, we apply a selection of analysis functions provided by  to particle-based simulations of the ABP model and to continuum simulations of the active model B+, respectively, and briefly discuss the results.







§ HOW TO USE 


 is designed with a user-focused mindset to simplify the access to simulation data within a few lines of Python code. Before discussing the general design of  and various examples on how to apply it to particle-based and continuum simulation data, we will demonstrate its general usage with a minimal example. For a quick start with , we recommend to download the demo files at <https://github.com/amepproject/amep/tree/main/examples> and to run them part by part while reading this section.



 §.§ Exemplary data: The active Brownian particle model


Within the following example, we load simulation data from a particle-based simulation of the overdamped active Brownian particle (ABP) model in two spatial dimensions. The ABP model is one of the simplest and most popular models to describe active particles that self propel in a certain direction which smoothly changes due to rotational diffusion <cit.>. Within this model, each particle is represented by a (slightly soft) disk/sphere of diameter σ, mass m, and moment of inertia I and possesses an inherent internal drive, denoted by an effective self-propulsion force F⃗_SP,i=γ_ t v_0 p̂(θ_i), where the self-propulsion direction is represented by p̂(θ_i)=(cosθ_i, sinθ_i) and v_0 denotes the terminal self-propulsion speed. The interactions between the particles are governed by an excluded-volume repulsive interaction potential u. The evolution of the particles' positions r⃗_i and orientations θ_i adheres to the Langevin equations <cit.>

    m ^2 r⃗_i/ t^2   = -γ_ tr⃗_i/ t - ∑_j=1
    j≠ i^N∇_r⃗_iu(r_ji) + γ_ t v_0 p̂(θ_i)
        + √(2 k_ B T_ bγ_ t)η⃗_i (t), 
     
    	I ^2 θ_i/ t^2   = -γ_ rθ_i/ t + √(2k_ B T_ bγ_ r)ξ_i(t).

Here, η⃗_i and ξ_i denote Gaussian white noise with zero-mean and unit variance, T_ b represents the bath temperature, γ_ t and γ_ r denote the translational and rotational drag coefficients, respectively, k_ B is the Boltzmann constant, and N denotes the total number of active particles. The overdamped limit can be obtained using m/γ_ t→ 0 and I/γ_ r→ 0 (see also Section <ref>).



 §.§ Installing 


 is a Python library that requires Python 3.10 or higher. To use , we recommend to install Python via Anaconda[See <https://docs.anaconda.com/free/anaconda/install/index.html> for instructions on how to install Anaconda.]. If Anaconda is installed, one can create a new environment, activate it, and install  through the terminal (Linux/macOS) or the Anaconda Prompt (Windows):
[language=bash, style=command] 
conda create -n amepenv python=3.10
conda activate amepenv
conda install conda-forge::amep

Now, one can start the Python interpreter to import and use :
[language=bash, style=command, firstnumber=4] 
python
>>> import amep
>>> 

Alternatively, we recommend to use Jupyter notebooks[For more information about Jupyter notebooks, see <https://jupyter.org/>.].



 §.§ Analyzing simulation data using 


Now, we use  to calculate the mean-square displacement (MSD), the diffusion coefficient, and the orientation autocorrelation function (OACF) of a single ABP and the radial distribution function (RDF) of multiple interacting ABPs. At the end of this section, we visualize the results in a combined plot. Exemplary code and data of the relevant ABP simulations is available at <https://github.com/amepproject/amep/tree/main/examples>.

First, we import  and load the simulation data of non-interacting overdamped ABPs:

import amep
traj_nonint = amep.load.traj(
    "/path/to/non_interacting_ABPs",
    mode = "lammps"
)

The function amep.load.traj creates an HDF5 file () in the background that contains all the simulation data and returns a ParticleTrajectory object which allows to easily access the data for further processing. Since the data has been produced using LAMMPS <cit.>, we load it using mode = "lammps". To save useful information for long-term storage, we add the author to the trajectory object, which will save this information within the linked HDF5 file:
[firstnumber=6]
traj_nonint.add_author_info(
    "Lukas Hecht",
    "affiliation",
    "Technical University of Darmstadt"
)
traj_nonint.add_author_info(
    "Lukas Hecht",
    "email",
    "lukas.hecht@pkm.tu-darmstadt.de"
)

This information can be accessed by calling traj_nonint.get_author_info("Lukas Hecht"), returning

'affiliation': 'Technical University of Darmstadt',
 'email': 'lukas.hecht@pkm.tu-darmstadt.de'

while traj_nonint.authors returns a list of all authors.  provides several methods to add more information to trajectory objects (ParticleTrajectory or FieldTrajectory) such as software information, simulation scripts, log files, or simulation parameters for a comprehensive and reproducible set of information about the simulation (see online documentation available at <https://amepproject.de> and Fig. <ref>).



Next, we calculate the MSD, which is defined as

    MSD(t) = 1/N∑_i=1^N[r⃗_i(t)-r⃗_i(0)]^2,

where r⃗_i(0), r⃗_i(t) denote the position of particle i at time 0 and t≥ 0, respectively. For a single ABP, it can be shown analytically that the MSD can be written as

    MSD(t)   = 2l_p^2(tτ_p -1 + e^-t/τ_p)+4D_ tt
        4D_ efft

with effective diffusion coefficient D_ eff=D_ t+l_ p^2/(2τ_ p) <cit.>. Here, τ_ p=1/D_ r denotes the persistence time, which signifies the time after which the directed motion of an ABP is randomized due to rotational diffusion, and l_ p=v_0τ_ p denotes the persistence length, which is the distance an ABP moves on average before its direction of motion is randomized. D_ t=k_ BT_ b/γ_ t and D_ r=k_ BT_ b/γ_ r are the translational and rotational diffusion coefficients, respectively, and v_0 is the terminal self-propulsion speed of the ABP. To calculate the MSD with , we create a MSD object via
[firstnumber=16]
msd = amep.evaluate.MSD(
    traj_nonint
)

which calculates the MSD for all frames and performs the average over all particles. The MSD object contains all information about the MSD and we can access the times via msd.times and the value for each individual frame via msd.frames for example. The returned objects are NumPy arrays and can therefore easily be used also elsewhere in Python.

To get the effective diffusion coefficient D_ eff from the MSD in the late time limit, we define the fit function f with the fit parameters as keyword arguments and create a corresponding Fit object:
[firstnumber=19]
def f(t, D = 1.0):
    return 4*D*t
Dfit = amep.functions.Fit(f)

By calling the object's fit method, which uses the scipy.odr package in the background, with the data to fit and an initial guess (using the p0 keyword), we obtain the optimal parameters. Here, we only want to fit the long-time behaviour, and hence, we only consider data for t>10^1τ_ p. The optimal parameters and their errors can then be retrieved from the Fit object:
[firstnumber=22]
mask = msd.times > 1e1
Dfit.fit(
    msd.times[mask],
    msd.frames[mask],
    p0 = [900]
)
print(
    f"D=Dfit.params[0]:0.2f, "    f"D-error=Dfit.errors[0]:0.2f"
)


D=885.32, D-error=0.59

This is fairly close to the expected value of D_ eff=883 as obtained from Eq. (<ref>) for ABPs with l_ p=1.0, τ_ p=1.0, v_0=42, and D_ t=1.0 as used in this example.

Next, we use the same simulation data to calculate and fit the OACF, which is given by 

    ⟨p̂(0)·p̂(t) ⟩ = 1/N∑_i=1^Np̂_i(0)·p̂_i(t)

and is equal to e^-D_ rt for overdamped ABPs <cit.>. Here, p̂_i is the orientation vector of the effective self-propulsion force of particle i. Again, by creating the corresponding evaluate and fit objects, we can analyze the OACF in a few lines of Python code. For the correct normalization, we specify the plane in which the OACF is to be calculated with direction = "xy":
[firstnumber=32]
oacf = amep.evaluate.OACF(
    traj_nonint,
    direction = "xy"
)
def f(t, Dr = 1.0):
    return np.exp(-Dr*t)
Tfit = amep.functions.Fit(f)
Tfit.fit(oacf.times, oacf.frames)
print(
    f"Dr=Tfit.params[0]:0.4f, "    f"Dr-error=Tfit.errors[0]:0.4f"
)


Dr=0.9890, Dr-error=0.0021

Again, we obtain a value close to the expected one of D_ r=1.0.

Next, we calculate the RDF. For that, we load simulation data of N=10^5 interacting ABPs moving in a two-dimensional periodic simulation box with a total packing fraction of φ=0.5 and create a RDF evaluate object. Note that a corresponding exemplary dataset is available at <https://github.com/amepproject/amep/tree/main/examples>. Here, we only want to consider frames that are in the steady state. Therefore, we skip the first 90 % of frames using skip = 0.9. Additionally, we specify the total number of frames to average over (time average) equally spaced in time over the last 90 % of the trajectory using the nav keyword, which is inherited from the BaseEvaluation class and is short for number of averages. Due to the large number of particles, it is worth to use multiprocessing, to set a maximum distance until which the RDF is calculated, and to use a large number of bins using the njobs, rmax, and nbins keywords, respectively:
[firstnumber=44]
traj_int = amep.load.traj(
    "/path/to/interacting_ABPs",
    mode = "lammps"
)
rdf = amep.evaluate.RDF(
    traj_int,
    skip = 0.9,
    nav = 100,
    nbins = 20000,
    rmax = 300,
    njobs = 128
)


Before visualizing the results, we save them in HDF5 files. For that,  offers two possibilities: One can store the result of either a single evaluate object (as done for the RDF) or multiple evaluate objects (as done for the MSD and the OACF) in one HDF5 file:
[firstnumber=56]
msd.save(
    "nonint-results.h5",
    database = True
)
oacf.save(
    "nonint-results.h5",
    database = True
)
rdf.save(
    "rdf.h5"
)

These can later be loaded by calling msd = amep.load.evaluation("nonint-results.h5", database = True).msd and rdf = amep.load.evaluation("rdf.h5") for example.

Finally, the data and fits can be visualized with the Matplotlib wrappers provided by  (Fig. <ref>):
[firstnumber=67]
fig, axs = amep.plot.new(
    (6.5,2), ncols = 3, wspace = 0.1
)
# plot msd
axs[0].plot(
    msd.times, msd.frames,
    "r-", label = "data"
)
axs[0].plot(
    msd.times, Dfit.generate(msd.times),
    "k–", label = "fit"
)
axs[0].set_xlabel("Time")
axs[0].set_ylabel("MSD")
axs[0].loglog()
axs[0].legend()
	
# plot oacf
axs[1].plot(
    oacf.times, oacf.frames,
    "r-", label = "data"
)
axs[1].plot(
    oacf.times, Tfit.generate(oacf.times),
    "k–", label = "fit"
)
axs[1].set_xlabel("Time")
axs[1].set_ylabel("OACF")
axs[1].semilogx()
axs[1].axhline(1/np.e, c = "k")
axs[1].axvline(1, c = "k")
axs[1].legend()
	
# plot rdf
axs[2].plot(rdf.r, rdf.avg, "r-")
axs[2].semilogx()
axs[2].set_xlabel("Distance")
axs[2].set_ylabel("RDF")
	
# plot boxes
amep.plot.draw_box(
    fig, [0, 0, 0.68, 1.01],
    edgecolor = "tab:blue", linestyle = "–",
    text = "Single ABP", c = "tab:blue"
)
amep.plot.draw_box(
    fig, [0.69, 0, 0.31, 1.01],
    edgecolor = "tab:orange", linestyle = "–",
    text = "Interacting ABPs", c = "tab:orange"
)	
# save figure in file
fig.savefig("how_to_use_amep.pdf")

The whole workflow of this minimal example is marked with red arrows in the flowchart shown in Fig. <ref> and its code is available at <https://github.com/amepproject/amep/tree/main/examples>.









§ DESIGN


In the previous section, we introduced a short example of how to use . We will now give an overview of the design of the  package. A more detailed description can be found in the online documentation available at <https://amepproject.de>.




 §.§ Modules


 contains a plethora of pre-implemented functions to analyze particle-based and continuum simulation data. In Tab. <ref>, the modules are listed with a short description and a few exemplary functions. The functions contained in these modules take NumPy arrays and native Python data types as input and therefore, can also be used without the data-handling framework provided by . In addition to the modules shown in Tab. <ref>,  has a plot module to visualize simulation data and results as well as an evaluate module, which provides various evaluation methods that use a full trajectory as input data. The plot and the evaluate module will be described in more detail below.






 §.§ Visualization


For visualizing simulation data and analysis results,  provides the plot module, which is a wrapper for the Matplotlib library <cit.>. The plot module includes functions to plot particles and continuum fields (cf. Figs. <ref> and <ref>), to create insets and colorbars (cf. Fig. <ref>), and to animate plots and simulation data (cf. Movies S1–S3, Supplemental Material). For example, a video of a trajectory traj can be created via traj.animate("video.mp4"). By varying the file extension, also other data formats can be used such as GIFs. Additionally, the plot module provides useful functions to format the axis of a figure. All visualization functions are optimized to create figures and videos that can be directly used for a publication in various journals. All figures within this paper with the exception of Fig. <ref> are created with .





 §.§ Evaluate module


We already introduced the different modules of , which contain a plethora of functions relevant for the analysis of active matter simulation data. The evaluate module is a special module in the sense that it uses functions from all other modules to calculate certain observables for a whole trajectory. In our first example in Section <ref>, we already used the MSD, OACF, and RDF classes, which are part of the evaluate module. We will now introduce the concept of these classes, which will be referred to as “evaluate classes” in the following. 

While the functions of the modules discussed above take NumPy arrays as input data, evaluate classes take a whole trajectory as an input. One example is the amep.evaluate.RDF class used in Section <ref>. To initialize an object of an evaluate class, the trajectory as well as certain parameters are supplied. The evaluate classes offer the functionality to average over multiple frames automatically (the keyword nav, which is an abbreviation for “number of averages”, allows to specify the number of frames to average over) as well as to skip a certain amount of frames at the beginning of the trajectory (in percent) by specifying the skip keyword. The resulting data can be returned as NumPy arrays. Furthermore, one or more evaluate objects can be stored in HDF5 files using their .save("filename.h5") method and can be read again with amep.load.evaluation("filename.h5"). The evaluate classes are a crucial part in the user-focused design of  and make it possible to achieve results within a few lines of code.





 §.§ File format


Based on the Hierarchical Data Format version 5 (HDF5) <cit.>,  introduces a new file format  to store simulation data and additional metadata. This format is used in the backend of . The HDF5 files are structured into groups and datasets. The  format has the following groups, subgroups, and attributes:



The group  contains information about the  version that has been used to create the  file. The group  contains the saved information about authors (cf. Section <ref>) and software. The  group gives the possibility to save text files such as simulation scripts and log files that correspond to the simulation data. In the  group,  stores parameters such as the simulation timestep for example. Additional simulation parameters can be added. The attribute  contains a flag about the type of data stored in the  file. This can be either "particle" or "field". The groups  and  contain user-defined information about the particles and the fields used in the simulation, respectively. Finally, the group  contains multiple datasets and subgroups with the simulation data. The dataset  stores a list of all the frame numbers (i.e., the number of timesteps for each frame) and the dataset  the corresponding (physical) time, while the individual frames of the simulation are stored in subgroups of  named by their simulation step. Within an individual frame, the simulation data is stored, e.g., coordinates and velocities for particle-based simulations or density and concentrations for continuum simulations, as separate datasets.







 §.§ Trajectories and frames


The  file is connected to Python via the amep.load module with which the simulation data is imported to a ParticleTrajectory or FieldTrajectory object. Saved evaluation objects can also be imported with the amep.load module. Because a ParticleTrajectory or FieldTrajectory object works as a reader for the  file, the simulation data (coordinates, velocities, density, concentration,...) is not stored in the main memory all the time but only the portion that is requested for the specific analysis. The simulation data can be accessed via BaseFrame or BaseField objects for particle-based and continuum data, respectively, which will be referred to as “frame” in the following. Technically, the ParticleTrajectory or FieldTrajectory object acts as a list of frames, i.e., frame = traj[0] returns the first frame of the trajectory traj. The data of one frame can be accessed through various methods. For example, the coordinates and velocities of particles can be accessed via frame.coords() and frame.velocities(). Other data can be accessed via the frame.data method, e.g., frame.data("mass") returns the mass of each particle and frame.data("rho") the density field of a continuum simulation. A list of all available keys can be accessed via frame.keys. All these methods return NumPy arrays for efficient calculations and simple integrity to other Python packages (see also Fig. <ref>).








§ ANALYZING PARTICLE-BASED SIMULATION DATA WITH 


We will now use  to analyze simulation data of large-scale simulations of ABPs. After introducing the simulation details, we demonstrate how to calculate certain observables using  and briefly discuss the results. The code examples demonstrate the workflow of  and serve as a guide for using . Further observables that are available in  but which are not discussed in this section are exemplarily shown in Fig. <ref>.



 §.§ Brownian dynamics simulations of active Brownian particles


As a first detailed example, we analyze particle-based simulation data obtained with the active Brownian particle (ABP) model as introduced in Section <ref> as Eqs. (<ref>) and (<ref>). The repulsive interaction is modeled by the Weeks-Chandler-Anderson (WCA) potential u(r_ji)=4ϵ[(σ/r_ji)^12 -(σ/r_ji)^6]+ϵ, where r_ji=|r⃗_j-r⃗_i| and u=0 for r_ji/σ≥ 2^1/6 <cit.>. For the simulations here, we have chosen l_ p=100σ, ϵ/k_ BT_ b=1.0, and for simplicity, γ_ t=γ_ r/σ^2 unless otherwise stated (note that in experiments of active granulates, the Stokes-Einstein relation does not apply <cit.>). Here, D_ t=k_ BT_ b/γ_ t and D_ r=k_ BT_ b/γ_ r are the translational and rotational diffusion coefficients, respectively. We define the Péclet number as Pe=v_0/√(2D_ tD_ r), which quantifies the strength of self-propulsion relative to diffusive motion. Note that all simulations are done in the overdamped regime, i.e., we choose a small mass m/(γ_ tτ_ p)=5× 10^-5. Since it has been found in previous works that the influence of the moment of inertia I on the simulation results is rather unimportant <cit.>, it is held constant at I/(γ_ rτ_ p)=0.33 unless otherwise stated. All simulations are done within a two-dimensional quadratic box of length L with periodic boundary conditions and up to N=1.28 × 10^6 particles with a time step of Δ t/τ_ p=10^-5–10^-6 using LAMMPS <cit.>. We choose Pe=70.7 and a packing fraction of φ=0.5 with φ=Nπσ^2/(4L^2) and start each simulation from uniformly distributed particle positions unless otherwise stated.

 can load the common LAMMPS plain text format in which LAMMPS stores one snapshot of the simulation per text file named as  for example, where the  is a placeholder for the current time step. These files are formatted as follows:



Here,  denotes a uniqe identifier for each particle,  denotes the particle type,  denote the position of each particle,  denote the forces acting on each particle,  are the components of the orientation vector p̂,  denotes the radius σ/2 of each particle, and  their mass m. Note that  is zero for all particles because the simulation is done in two spatial dimensions. An exemplary dataset is available at <https://github.com/amepproject/amep/tree/main/examples>.  reads all data from each text file and converts it into the  format. Each item from the  section in the LAMMPS text file can then be accessed with the .data() method of the corresponding BaseFrame object, e.g., the force in x direction of frame 5 can be returned as NumPy array by calling 

import amep
traj = amep.load.traj(
    "/path/to/data",
    mode = "lammps"
)
fx = traj[5].data("fx")

For many standard quantities such as coordinates, forces, or velocities,  provides further commands exemplarily shown below:[See online documentation at <https://amepproject.de> for further details.]
[firstnumber=7]
coords = traj[5].coords()
forces = traj[5].forces()
mu = traj[5].data("mux", "muy")






 §.§ Motility-induced phase separation


For large enough packing fraction and large enough Pe, ABPs can phase separate into a dense and a dilute phase. This phenomenon is well-known as motility-induced phase separation (MIPS) <cit.> and can be understood as follows: If two ABPs collide, they can block each other due to their effective self-propulsion force. The collision can be resolved if the ABPs reorient their self-propulsion direction due to rotational diffusion and the self-propulsion direction of one ABP deviates from the other. Small clusters can form if more ABPs collide with the two ABPs blocking each other. Roughly, if now the mean time τ_ c between collisions is smaller than the mean time τ_ p a particle randomly reorients its self-propulsion direction, small clusters tend to grow, which finally results in a phase separation where an active gas coexists with a dense liquid <cit.>. This process is exemplarily visualized by the snapshots shown in Fig. <ref>, which have been created using the amep.plot.particles function. In the following, we will mainly focus on the analysis of the simulation visualized in Fig. <ref>.




 §.§ Voronoi diagrams and local order


Whether a simulation undergoes MIPS can be quantified by calculating the distribution of the local density or local area fraction, which is unimodal in the uniform regime and becomes bimodal in the MIPS regime <cit.>.  provides multiple functions to calculate the local density or area fraction: amep.order.local_number_density, amep.order.local_mass_density, and amep.order.local_packing_fraction determine the local number density, mass density, and area/volume fraction, respectively, from averages over circles of radius R, which can be written as φ_ loc(r⃗_i)=∑_j=1^Nσ_j^2 H(R-|r⃗_i-r⃗_j|)/(4 R^2) in case of the local area fraction. Here, H is the Heaviside step function and σ_j is the diameter of particle j. Alternatively, amep.order.voronoi_density calculates the local mass density, number density, or area/volume fraction based on a Voronoi tesselation. Here, we will use the latter for calculating the distribution of (i) the local area fraction and (ii) the number of next neighbors. 

The Voronoi diagram serves as a method for modeling the structures of materials across various disciplines, including crystallography, ecology, astronomy, epidemiology, geophysics, computer graphics, and more <cit.>. In essence, when given a set of points in the plane, the Voronoi diagram divides the plane based on the nearest neighbor rule, associating each point with the region of the plane closest to it. The mathematical definition extends to N-dimensions, often referred to as Voronoi tessellation <cit.>. For a set of finite points P = {p_1,p_2, ⋯, p_n } in a subspace ζ in the Euclidean space, the Voronoi cell is formed through the division of the plane into regions, where each region encompasses all points that are closer to each p_i. Points equidistant from at least two points in set P are not part of the Voronoi cell. Instead, they constitute the Voronoi edges. The compilation of all Voronoi cells forms the Voronoi tessellation associated with ζ.



The Voronoi tesselation is included in the order module of  as function amep.order.voronoi. It calculates the Voronoi diagram for a given set of particle coordinates using the scipy.spatial.Voronoi class of the SciPy package <cit.>. Here, we use the Voronoi tesselation to calculate the number of next neighbors and the local area fraction. Let us first calculate the Voronoi diagram of the last frame (traj[-1]) via

import amep
# load the simulation data
traj = amep.load.traj(
    "/path/to/data",
    mode = "lammps"
)
# calculate the Voronoi diagram
vor, ids = amep.order.voronoi(
    traj[-1].coords(),
    traj[-1].box,
    pbc = True
)

The obtained Voronoi diagram is demonstrated in Fig. <ref>a for the simulation snapshot shown in Fig. <ref>c. It can be easily visualized using amep.plot.voronoi. Next, we calculate the number of next neighbors from the Voronoi diagram and its distribution via
[firstnumber=13]
# number of next neighbors for each particle
nnn, _, _, _, _ = amep.order.next_neighbors(
    traj[-1].coords(),
    traj[-1].box,
    vor = vor,
    ids = ids,
    pbc = True
)
# distribution of the number of next neighbors
nndist, bins = amep.statistics.distribution(
    nnn
)

As we can see in Fig. <ref>b, the distribution of the number of next neighbors attains its highest point at N_ nn=6. This observation indicates the existence of a hexagonal structure within the dense phase, as we will discuss in more detail below. Finally, we calculate the local area fraction using the Voronoi areas and evaluate its distribution:
[firstnumber=25]
# local area fraction of each particle
ld = amep.order.voronoi_density(
    traj[-1].coords(),
    traj[-1].box,
    radius = traj[-1].radius(),
    mode = "packing",
    vor = vor,
    ids = ids
)
# distribution of the local area fraction
lddist, bins = amep.statistics.distribution(
    ld
)

If one would like to calculate the distribution of the local area fraction averaged over several frames of a trajectory, i.e., using a time average, one can simply create an amep.evaluate.LDdist object which is directly performing the time average and allows us to store the results in an HDF5 file. Additionally, multiple results can be stored in the same HDF5 file using 's database feature: As an example, we calculate the distribution of the local area fraction for five independent simulation runs (ensembles) stored in directories – and store them together in one HDF5 file :

import amep
# load all trajectories
trjs = [
    amep.load.traj(
        f"/path/to/data/doi:02",
        mode = "lammps"
    ) for i in range(5)
]
for i,traj in enumerate(trjs):
    # distribution of the local area fraction
    ld = amep.evaluate.LDdist(
        traj,
        nav = traj.nframes,
        use_voronoi = True,
        mode = "packing"
    )
    # set name under which the data
    # is stored in the HDF5 file
    ld.name = f"doi:02"

    # save all in file ld.h5
    ld.save("ld.h5", database = True)

Now, we can calculate the ensemble average:
[firstnumber=23]
# load the analysis results
lds = amep.load.evaluation(
    "ld.h5", database = True
)
# ensemble average
ensavg = 0.0
for i in range(5):
    ensavg += lds[f"doi:02"].avg/5

The ensemble-averaged result is shown in Fig. <ref>c. As expected, the distribution of the local area fraction shows two peaks indicating that the system is phase-separated into a dense and a dilute phase. Note that the high-density peak is located at φ_ loc>1.0 due to the softness of the particles.



 §.§ Hexagonal order




In the last paragraph, we saw that on average, each particle has six next neighbors, indicating that a hexagonal order dominates the local order of the particles. To quantify the local order in more detail, we calculate the hexagonal order parameter

    ψ_6(r⃗_i)=1/6∑_j=1^6exp{ i 6θ_ij}

of each particle, where the sum goes over the six nearest neighbors of particle i and θ_ij is the angle between the connection line from r⃗_i to r⃗_j and the x axis <cit.>. In two spatial dimensions, the particles within the dense phase typically show local hexagonal order while they are disordered in the dilute phase. Therefore, the distribution of ψ_6 can also be used to assess whether a system is phase separated similarly to the local area fraction <cit.>. Calculating the distribution of the hexagonal order parameter can be easily done using 's evaluate module:

import amep
# load the simulation data
traj = amep.load.traj(
    "/path/to/data",
    mode = "lammps"
)
# psi6 distribution (all frames)
p6dist = amep.evaluate.Psi6dist(
    traj, 
    nav = traj.nframes
)
# save results in HDF5 file
p6dist.save("p6dist.h5")

# plot the result of the last frame
fig, ax = amep.plot.new()
ax.plot(
    p6dist.psi6,
    p6dist.frames[-1,0]
)
fig.savefig("p6dist.png")

The result is demonstrated in Fig. <ref>f, where we have averaged over five independent ensembles. The broad peak at small values corresponds to the dilute phase while the sharp peak at |ψ_6|=1 corresponds to the hexagonally packed dense phase as shown in Fig. <ref>e. It is also common to analyze the spatial range of the hexagonal order to check whether there exists a short-range or (quasi) long-range hexagonal order like in a crystal. To this end, one can calculate the spatial correlation function of the hexagonal order parameter

    g_6(r) = ⟨ψ_6^*(r⃗_i)ψ_6(r⃗_j)⟩/⟨|ψ_6(r⃗_i)|^2⟩

with r=|r⃗_i-r⃗_j| <cit.>. Here, we calculate g_6(r) for ABP simulations  with ϵ/(k_ BT_ b)=10, N=100 000 particles, and at φ=0.8 starting from a hexagonal lattice instead of a uniform distribution and average over multiple frames in the steady state:

import amep
# load simulation data
traj = amep.load.traj(
    "/path/to/data",
    mode = "lammps"
)
# calculate g6
g6 = amep.evaluate.HexOrderCor(
    traj,
    nav = traj.nframes,
    njobs = 16,
    rmax = 100.0,
    skip = 0.5
)
# save results in an HDF5 file
g6.save(
    "g6.h5"
)
# plot average
fig, ax = amep.plot.new()
ax.plot(
    g6.r,
    g6.avg
)
fig.savefig("g6.png")

Since this calculation is computationally expensive, we speed it up using parallelization by specifying the number of jobs that should run in parallel with the njobs keyword. Additionally, we give a maximum distance up to which the correlation function should be calculated and we skip the first 50% of the trajectory to ensure that only the second half of the trajectory, which is in the steady state, is used for the calculation. The result is shown in Fig. <ref>d together with the corresponding snapshots in Fig. <ref>a–c for three different Pe. In accordance to Ref. <cit.>, we observe a constant g_6 at small Pe and an algebraic decay of the form r^-ν with exponent ν at higher Pe.



 §.§ Structure factor and coarsening


To further probe the order of an active system, one can exploit the radial distribution function g(r), which we have exemplarily calculated in Section <ref> using amep.evaluate.RDF. From an experimental viewpoint, it is easier to obtain the structure factor S(q⃗,t), which is defined as <cit.>

    S(q⃗,t)=1/N⟨∑_i=1^N∑_j=1^Nexp{ iq⃗·[r⃗_i(t) - r⃗_j(t)] }⟩.

Here, q⃗ represents the wave vector, r⃗_i(t) and r⃗_j(t) denote the positions of particles i and j at time t, respectively, and N is the total number of particles. For isotropic systems, the structure factor only depends on the magnitude q=|q⃗| of the wave vector. It gives information about the density response to an external perturbation of wave length 2π/q and can be probed experimentally via scattering experiments. Note that S(q⃗,t) is directly related to  g(r⃗,t) via Fourier transform. However, in practice, it is often preferable to calculate S(q⃗, t) directly to achieve good numerical results in the low q regime. Within , S(q⃗,t) can be directly calculated with amep.evaluate.SF2d and the isotropic structure factor S(q,t) with q=|q⃗| (i.e., S(q⃗,t) averaged over the direction of q⃗) with amep.evaluate.SFiso. Here, we use S(q,t) to analyze the coarsening process of the clusters forming when the ABPs undergo MIPS. To this end, we deduce a characteristic length scale L(t) based on the first moment of S(q,t) given by <cit.>

    L(t) =∫_2π/L^q_cut S(q,t)  q/∫_2π/L^q_cut q S(q,t)  q,

where L is the length of the simulation box. We set the upper limit q_ cutσ=0.3, i.e., we only consider length scales larger than 2π/q_ cut≈ 21σ. Notably, under suitable conditions, overdamped ABPs undergoing MIPS exhibit an effective mapping onto a suitable equilibrium system at coarse-grained scales, as established in the literature <cit.>. This mapping explains why the coarsening dynamics follows the universal law L(t) ∼ t^1/3 – a characteristic scaling behavior observed in equilibrium systems.



To visually elucidate this process, we present instances of S(q,t) at distinct times t/τ_p=0.1, 4.2, 800.0 (Fig. <ref>a) corresponding to the snapshots shown in Fig. <ref>. These results can be obtained using the following code example:

import amep
import numpy as np
# load simulation data
traj = amep.load.traj(
    "/path/to/data",
    mode = "lammps"
)
# calculate S(q)
sf = amep.evaluate.SFiso(
    traj,
    nav = traj.nframes
)
# plot S(q) of the last frame
fig, axs = amep.plot.new()
axs.plot(sf.q, sf.frames[-1][0])
fig.savefig("sf.pdf")

We can now subsequently compute L(t) by using the amep.utils.domain_length function and fit a power-law of the form L(t)=L_0t^α to it using amep.functions.Fit:
[firstnumber=17]
# calculate domain length
L = np.zeros(len(sf.frames))
for i,frame in enumerate(sf.frames):
    length = amep.utils.domain_length(
        frame[0], sf.q, qmax = 0.3
    )
    L[i] = length

# fit growth exponent
def f(x, L0 = 1, alpha = 1):
    return L0*x**alpha
fit = amep.functions.Fit(f)
mask = sf.times > 10
fit.fit(sf.times[mask], L[mask])

# plot data and fit
fig, axs = amep.plot.new()
axs.plot(sf.times, L, label = "data")
axs.plot(
    sf.times[mask],
    fit.generate(sf.times[mask]),
    marker = "", ls = "–",
    label = "fit"
)
axs.legend()
fig.savefig("domain-length.pdf")

The result is illustrated in Fig. <ref>b and we obtain a growth exponent of α=0.324± 0.001 in accordance with the literature <cit.>.



 §.§ Cluster analysis


Finally, we analyze the clustering of ABPs when undergoing MIPS by using 's cluster module. For particle-based simulation data, we define a cluster as a collection of particles in which each particle has an interparticle distance smaller than r_ max to some other particle, where the cut-off distance r_ max can be chosen by the user. Particle i and j with position coordinates r⃗_i and r⃗_j, respectively, belong to the same cluster if |r⃗_i-r⃗_j | ≤ r_ max. To find the distance between different particle pairs and identify the neighboring particle pairs that satisfy this distance criterion, we use the KDTree algorithm (scipy.spatial.KDTree) from the SciPy library <cit.>. In the following, we will first show how to identify clusters using  and afterwards analyze the coarsening of the clusters as well as their radius of gyration and fractal dimension. 

Let us consider the last frame of a simulation showing MIPS (Fig. <ref>c). The clusters can be identified using the amep.cluster.identify function: 

import amep
import numpy as np
# load simulation data
traj = amep.load.traj(
    "/path/to/data",
    mode = "lammps"
)
# get the last frame
frame = traj[-1]

# cluster detection
clusters, idx = amep.cluster.identify(
    frame.coords(),
    frame.box,
    pbc = True,
    rmax = 1.122
)

It takes the coordinates of all particles (frame.coords()) and the boundaries of the simulation box (frame.box) as an input and returns a list (clusters) of all clusters, sorted in descending order of their size, with each list element containing the indices of the particles belonging to the respective cluster. It also returns the array idx which stores the cluster indices assigned to each particle. The optional argument pbc is used to consider periodic boundary conditions and the cut-off distance rmax is set to the cut-off distance of the WCA potential (i.e., the contact distance). The algorithm works not only with simulation data of particles of the same size but can also identify clusters comprising particles of different sizes, which requires different values of the cut-off distance r_ max depending on the pair of particles. In the latter case, the user can provide the sizes of the particles (diameter) through the additional keyword sizes in amep.cluster.identify. In Fig. <ref>a, we show the same snapshot as in Fig. <ref>c, but with the particles of the eight largest clusters colored according to their cluster ids.

Based on the identified clusters, one can calculate various properties of the different clusters using the cluster module. One of them is the cluster size, defined as the number of particles within a cluster, the distribution of which often provides an insight into coexisting phases in a system <cit.>. We can calculate the cluster sizes and masses by using the amep.cluster.sizes and amep.cluster.masses functions:
[firstnumber=18]
sizes = amep.cluster.sizes(
    clusters
)
masses = amep.cluster.masses(
    clusters, frame.data("mass")
)

Based on sizes, we can now calculate the cluster size distribution p(s)=N_s/(∑_s N_s), where N_s is the number of clusters with size s. Since we are interested in the coarsening behavior of the large clusters but their number is relatively small, it has been proven to be beneficial to analyze the weighted cluster size distribution

    p_ w(s)=sN_s/∑_s sN_s,

where each cluster is weighted with its size s <cit.>. Here, we use the amep.statistics.distribution function with logarithmic bins:
[firstnumber=24]
# weighted cluster size distribution
hist, bins = amep.statistics.distribution(
    sizes, logbins = True,
    nbins = 50, density = False
)
hist = hist*bins/np.sum(hist*bins)

# plot result
fig, axs = amep.plot.new()
axs.bar(
    bins, hist, width = 0.2*bins
)
axs.loglog()
fig.savefig("size-dist.pdf")

The result is shown in Fig. <ref>b and exhibits two distinct peaks at small and large cluster sizes suggesting that the system is phase separated into two distinct phases. Large clusters characterize the dense liquid-like phase whereas the dilute gas-like phase comprises smaller clusters <cit.>.



The cluster sizes can also be used to study the average growth rate of the clusters, which is a commonly used measure of the coarsening kinetics of phases in soft matter and biophysics <cit.>. Here, we use amep.evaluate.ClusterGrowth to calculate the weighted mean cluster size ⟨ s⟩_ w and the mean cluster size ⟨ s⟩ defined as

    ⟨ s⟩_ w   =∑_s sp_ w(s)=∑_s s^2N_s/∑_s sN_s
    ⟨ s⟩   =∑_s sp(s)=∑_s sN_s/∑_s N_s

over time (Fig. <ref>c). Similar to the domain length L(t) obtained from the structure factor above, we will deduce a growth exponent from the growing mean cluster sizes. Note that for L(t), we only considered qσ <q_ maxσ=0.3, i.e., length scales larger than l_ min=2π/q_ max≈ 21σ, which resulted in a growth exponent of α≈ 1/3 (where L(t)∝ t^α). Since the cluster size measures the number of particles within a cluster, which is proportional to the area of the cluster, we can define a cluster length as l/σ=√(⟨ s⟩) that is expected to also scale as l∝ t^1/3. However, this holds true only if we either disregard very small clusters that do not grow over time or assign greater importance to larger clusters by calculating the weighted mean cluster size. Therefore, additionally to the weighted mean cluster size, we also compute the mean cluster size considering all the clusters (i.e., having at least two particles) and only the large clusters with at least 400 particles (in accordance to l_ min≈ 21σ used for the calculation of L(t)).
[firstnumber=38]
# mean cluster size (>2)
mean_2 = amep.evaluate.ClusterGrowth(
    traj,
    mode = "mean",
    min_size = 2
)
# mean cluster size (>400)
mean_400 = amep.evaluate.ClusterGrowth(
    traj,
    mode = "mean",
    min_size = 400
)
# weighted mean cluster size
weighted_mean = amep.evaluate.ClusterGrowth(
    traj,
    mode = "weighted mean"
)

We now use amep.functions.Fit to obtain the growth exponent from l(t)∝ t^β (here, exemplarily done for the weighted mean in the logarithmic domain):
[firstnumber=55]
# define fit function
def f(l, beta = 1.0, l0 = 1.0):
    return np.log(l0) + beta*l

# create fit object
fit = amep.functions.Fit(f)

# fit function to data at large times
# in logarithmic domain
mask = weighted_mean.times > 3e1
fit.fit(
    np.log(weighted_mean.times[mask]),
    np.log(weighted_mean.frames[mask])/2
)
print(fit.results)

# plot data and fit
fig, axs = amep.plot.new()
axs.plot(
    weighted_mean.times,
    np.sqrt(weighted_mean.frames)
)
axs.plot(
    weighted_mean.times[mask],
    np.exp(fit.generate(
        np.log(weighted_mean.times[mask])
    ))
)
axs.loglog()
fig.savefig("cluster-growth.pdf")


'beta': (0.297, 0.004),
 'l0': (64.6, 1.1)

The results are demonstrated in Fig. <ref>c. Consistent with the established ∝ t^1/3 growth of the characteristic domain length L(t) of such ABP clusters <cit.>, we obtain an exponent β≈ 1/3 for the cluster length calculated from the weighted mean and the mean over all clusters that are larger than 400 particles (Fig. <ref>c). For the mean over all clusters larger than two particles, the growth rate is significantly smaller because very small clusters do not grow.

To demonstrate the versatility of the cluster module, we present two additional calculable quantities using : the radius of gyration R_ g and the linear extension l_ e of the clusters. For a cluster composed of s particles of masses m_i, i=1,2, …, s, located at fixed distances d_i from the cluster's center of mass, the radius of gyration is defined as <cit.>

    R_ g=√(∑_i=1^s m_i d_i^2/∑_i=1^s m_i).

The linear extension (also known as the end-to-end length) of a cluster is defined as the maximal distance between two particles in the cluster, i.e,

    l_ e=max _{i, j}|r⃗_i-r⃗_j|,

with r⃗_i denoting the position vector of the i-th particle and i,j=1,2, …, s <cit.>. The mean linear extension along with the mean size of the largest cluster is commonly used as an order parameter to characterize isotropic percolation phase transitions in different systems <cit.>. On the other hand, the dependence of the radius of gyration on the cluster size s (or mass) allows us to understand the (possibly fractal <cit.>) geometry of the clusters and calculate their fractal dimension d_ f according to the relation R_g ∝ s^1/d_ f <cit.>. With , we can calculate such cluster properties easily by using its cluster module (here exemplarily done for the largest cluster):
[firstnumber=85]
# ids of particles in largest cluster
ids = clusters[0]

# center of mass
com = amep.cluster.center_of_mass(
    frame.coords()[ids],
    frame.box,
    frame.data("mass")[ids],
    pbc = True
)
# geometric center
gmc = amep.cluster.geometric_center(
    frame.coords()[ids],
    frame.box,
    pbc = True
)
# radius of gyration
rg = amep.cluster.radius_of_gyration(
    frame.coords()[ids],
    frame.box,
    frame.data("mass")[ids],
    pbc = True
)
# linear extension
le = amep.cluster.linear_extension(
    frame.coords()[ids],
    frame.box,
    frame.data("mass")[ids],
    pbc = True
)

To handle periodic boundary conditions,  uses the method proposed in Ref. <cit.>. The radius of gyration and the linear extension of the largest cluster are visualized in Fig. <ref>a. Additionally, we plotted R_ g and l_ e as function of the cluster size s in Fig. <ref>d to obtain the fractal dimension using amep.functions.Fit to fit the function R_ g(s)∝ s^1/d_ f. We obtain d_ f = 1.99± 0.03, i.e, essentially the same as the spatial dimension of the system, which is expected here since the clusters are compact.







§ ANALYZING CONTINUUM SIMULATION DATA WITH 


Let us now discuss some examples on how to analyze continuum simulation data with . To this end, we will use numerical solutions of the active model B+, which is a corresponding continuum model for active Brownian particles. We will first introduce the model and afterwards demonstrate and discuss certain observables and minimal code examples using .



 §.§ Active model B+


The active model B+ (AMB+) describes a system of active Brownian particles featuring a generic self-propulsion mechanism that can lead to interesting collective phenomena. It models the time evolution of the scalar field ϕ=(2ρ-ρ_H-ρ_L)/(ρ_H-ρ_L), where ρ_H and ρ_L denote the particle number density at the low-density and the high-density critical point, respectively <cit.>. The active model B+ is given by <cit.>

    ∂ϕ/∂ t=   -∇·[-M∇(δℱ/δϕ+λ|∇ϕ|^2).
       .+ ζ M(∇^2ϕ)∇ϕ + √(2k_ BTM)Λ⃗], 
    ℱ[ ϕ ] =   ∫d^3r [ a/2ϕ^2+b/4ϕ^4+K/2|∇ϕ|^2 ].

Here, the free-energy functional ℱ is approximated up to the order ϕ^4 and up to square-gradient terms, and Λ⃗(r⃗,t) denotes Gaussian white noise with zero mean and unit variance. The mobility is denoted by M, the temperature is given by k_ BT with Boltzmann constant k_ B, and a,b,K,ζ,λ are free parameters of the model. Here, we use a=-0.25, b=0.25, K=1.0, ζ=1.0, λ=-0.5, k_ BT=0.2, and M=1.0 corresponding to a parameter regime showing phase separation <cit.>. We start from a uniform initial condition with ϕ_0=-0.4 disturbed with weak fluctuations. Here, we used an in-house finite volume solver written in  to numerically integrate Eq. (<ref>) using a grid of 256× 256 grid points with a grid spacing of Δ x=Δ y = 1.0 and time step Δ t=0.001. All simulations run for 10^8 time steps. To load the resulting data with , the data has to be stored in a certain data format as discussed in Subsection <ref> and as exemplified in the exemplary dataset which can be downloaded from <https://github.com/amepproject/amep/tree/main/examples>.



 §.§ Motility-induced phase separation




Similar to the particle-based simulations of ABPs, the AMB+ also undergoes MIPS in a certain parameter regime <cit.>. Whether the system is phase separated can again be determined by evaluating the local density distribution. In case of continuum data, one can simply calculate the distribution of the density at each grid point, as done by the amep.evaluate.LDdist class:

import amep
# load the data
traj = amep.load.traj(
    "/path/to/data",
    mode = "field"
)
# local density distribution of each frame
lddist = amep.evaluate.LDdist(
    traj,
    nav = traj.nframes,
    xmin = -1.5,
    xmax = 1.5,
    ftype = "phi"
)
# save results in HDF5 file
lddist.save("lddist.h5")

# plot for the last frame
fig, axs = amep.plot.new()
axs.plot(lddist.ld, lddist.frames[-1,0])
axs.set_xlabel(r"ϕ_ loc")
axs.set_ylabel(r"p(ϕ_ loc)")
fig.savefig("lddist.pdf")

The results are demonstrated in Fig. <ref> at different times together with the corresponding snapshots. While the distribution is unimodal at the beginning of the simulations (which start with a uniform distribution), it becomes bimodal at long times showing that the system phase separates into a dense and a dilute phase by forming dense clusters that coexist with a surrounding gas-like phase.



 §.§ Coarsening processes


Let us now analyze the coarsening behavior of the clusters based on the isotropic structure factor. For a continuum field, the (two-dimensional) structure factor can directly be calculated from the numerical Fourier transform of the scalar field ϕ as <cit.>

    S(q⃗,t) = ⟨ϕ(q⃗,t)ϕ(-q⃗,t)⟩ = ⟨|ϕ(q⃗,t)|^2⟩

using the amep.continuum.sf2d function. Here, ϕ(q⃗,t) is the spatial Fourier transform of ϕ(r⃗,t). To obtain the isotropic structure factor, the resulting two-dimensional structure factor must be averaged over the direction of q⃗, i.e., (using polar coordinates)

    S(q,t) = 1/2π∫_0^2π dφ S((qcos(φ),qsin(φ)), t)

which can be done in  with the amep.utils.sq_from_sf2d function:

import amep
import numpy as np
# load data
traj = amep.load.traj(
    "/path/to/data",
    mode = "field"
)
# get last frame
frame = traj[-1]

# 2d structure factor
sf2d, qx, qy = amep.continuum.sf2d(
    frame.data("phi"),
    *frame.grid
)
# isotropic structure factor
sfiso, q = amep.utils.sq_from_sf2d(
    sf2d, qx, qy
)

's evaluate module allows to calculate the isotropic structure factor directly from the traj object and also performs a time average:
[firstnumber=20]
# isotropic structure factor
# for all frames (time average)
sf = amep.evaluate.SFiso(
    traj,
    nav = traj.nframes,
    ftype = "phi"
)

Here, we can specify which field of the given trajectory should be used via ftype. From the structure factor, we can then again calculate the domain length as defined in Eq. (<ref>) via
[firstnumber=27]
# domain length over time
L = np.zeros(traj.nframes)
for i,f in enumerate(sf.frames):
    L[i] = amep.utils.domain_length(
        f[0], f[1]
    )

Note that we do not specify qmax here because we integrate over all q up to its largest possible value q_ max=π/Δ x given by the grid spacing Δ x. The results are demonstrated in Fig. <ref>, which shows three exemplary curves of the structure factor at three different times and the domain length over time. From the domain length, we extract the growth exponent by fitting a power law to the domain length at long times in the logarithmic domain:
[firstnumber=33]
# define fit function
def f(t, L0 = 1.0, alpha = 1.0):
    return np.log(L0) + alpha*t
# create fit object
fit = amep.functions.Fit(f)

# fit at large times
mask = traj.times > 1e1
fit.fit(
    np.log(traj.times[mask]),
    np.log(L[mask])
)
print(fit.results)


'L0': (4.168, 0.085),
 'alpha': (0.2702, 0.0029)

Interestingly, for our continuum simulations of the active model B+, we obtain α≈ 0.27, which is smaller than the value α≈ 1/3 as obtained in our particle-based simulations. Such subdiffusive scaling has been obtained in previous studies of active field theories as well and is typically expected to be an intermediate scaling regime that leads to a t^1/3 scaling at longer times <cit.>.





 §.§ Cluster analysis


Next, we will analyze the clusters forming at late times in more detail by performing a similar cluster analysis as done in Subsection <ref>. For the continuum data, a cluster is defined as a connected region of similar and higher density than the surroundings.  provides two algorithms to detect clusters in continuum data (i.e., discretized density fields). The standard algorithm is dividing up pixels according to their values relative to a threshold value, i.e., for a scalar continuum field ϕ, a zero is assigned to all pixels with ϕ≤ a_ thres(ϕ_ max-ϕ_ min) and a one to all pixels with ϕ > a_ thres(ϕ_ max-ϕ_ min). Here, a_ thres∈[0,1] denotes the relative threshold and ϕ_ min, ϕ_ max are the minimum and maximum values of the continuum field. On the resulting two-valued image, all connected regions are then labeled as a cluster using skimage.measure.label <cit.>. Within , this algorithm is called "threshold". An alternative way for cluster detection is the watershed algorithm <cit.>. It is more stable against slowly varying background fields but needs more fine-tuning of parameters to work. Therefore, the watershed algorithm is not the first choice for detecting clusters over time, i.e., for multiple frames of a trajectory, because the parameters may need to be adjusted for each frame individually. In , the skimage.segmentation.watershed function is used for the "watershed" cluster detection <cit.>.

Clusters within a continuum field can be identified with  using the amep.continuum.identify_clusters function. In the following example, we detect the clusters in the last frame of a continuum simulation using the "threshold" method:

import amep
# load data
traj = amep.load.traj(
    "path/to/data",
    mode = "field"
)
# get the last frame
frame = traj[-1]

# identify clusters
ids, labels =amep.continuum.identify_clusters(
    frame.data("phi"),
    pbc = True,
    threshold = 0.1,
    method = "threshold"
)

Here, frame.data("phi") returns an array of values of field "phi" at each point of the underlying discretized grid, the pbc keyword can be set to True to apply periodic boundary conditions, and threshold specifies the relative threshold a_ thres. The keyword method chooses between the "threshold" or "watershed" method. The amep.continuum.identify_clusters function assigns a unique identifier to each detected cluster returned as NumPy array (ids in the example above), and it returns an array of the same shape as the underlying discretized grid denoting which grid point belongs to which cluster (labels in the example above). The result is examplarily visualized in Fig. <ref>a.

As a next step, the amep.continuum.cluster_properties function can be used to calculated certain properties of the detected clusters such as their sizes, masses, centers, or radii of gyration:
[firstnumber=18]
s, gmc, com, rg, le, gt, it =amep.continuum.cluster_properties(
    frame.data("phi"),
    *frame.grid,
    ids,
    labels,
    pbc = True
)

where s, gmc, com, rg, le, gt, it denote the size, geometric center, center of mass, radius of gyration, linear extension, gyration tensor, and inertia tensor of each cluster, respectively. Here, the size of cluster i within a field ϕ is defined as the integral over its area A_i, i.e., s_i=∫_A_i d^2r ϕ(r⃗). All other cluster properties are calculated with the same methods as used for the particle-based simulation data by treating each grid point (i,j) as a particle at position r⃗_ij with “mass” ϕ(r⃗_ij). The radius of gyration and the linear extension of the largest cluster are visualized in Fig. <ref>a.



Similarly to the cluster analysis for particle-based data, we can now calculate the weighted cluster-size distribution, the growth exponents, and the fractal dimension of the clusters exploiting 's evaluate module. Let us first calculate the weighted cluster-size distribution as defined in Eq. (<ref>):
[firstnumber=26]
# weighted cluster size distribution
cd = amep.evaluate.ClusterSizeDist(
    traj,
    nav = traj.nframes,
    ftype = "phi",
    method = "threshold",
    threshold = 0.1,
    use_density = False,
    nbins = 50,
    logbins = True,
    xmin = 1e0,
    xmax = 1e4
)

Here, the keyword use_density allows to specify whether the integrated density s_i (use_density = True) as defined above or area A_i (use_density = False) should be used as size of cluster i. Furthermore, we have used the threshold method and logarithmic bins and the result is shown in Fig. <ref>b. Next, we analyze the cluster growth by calculating the average cluster size over time using the weighted mean and mean as defined in Eqs. (<ref>) and (<ref>), respectively: 
[firstnumber=39]
# mean cluster sizes
mean = amep.evaluate.ClusterGrowth(
    traj,
    ftype = "phi",
    method = "threshold",
    mode = "mean",
    threshold = 0.1,
    use_density = False
)
# mean cluster size over all clusters
# with size larger than 20
mean_20 = amep.evaluate.ClusterGrowth(
    traj,
    ftype = "phi",
    method = "threshold",
    mode = "mean",
    threshold = 0.1,
    use_density = False,
    min_size = 20
)
# weighted mean cluster size
weighted_mean = amep.evaluate.ClusterGrowth(
    traj,
    ftype = "phi",
    method = "threshold",
    mode = "weighted mean",
    threshold = 0.1,
    use_density = False
)

Again, we define the cluster length as l=√(⟨ s⟩) and use amep.functions.Fit to obtain the growth exponent from l(t)∝ t^β as already demonstrated in Subsection <ref>. Consistent with the growth exponent α≈ 0.27 obtained from the domain length L(t) (see Subsection <ref>), we obtain β≈ 0.26 (Fig. <ref>c). Additionally, we plotted the radius of gyration R_ g as function of the cluster size s in Fig. <ref>d to obtain the fractal dimension using amep.functions.Fit to fit the function R_ g(s)∝ s^1/d_ f. We obtain d_ f = 1.99± 0.01, i.e, the same as the spatial dimension of the system, which is expected here since the clusters are compact.



 §.§ Continuum data format


Continuum simulation data as analyzed within this work has to be stored in a specific format such that it can be loaded with . In the following, we briefly introduce the basic format requirements. In , field data is internally stored using the  file format, similar to the particle-based simulation data. Converting field data to an  dataset is done by use of a reader class (amep.reader.ContinuumReader). This data reader expects the following format: The standard file structure for field data is inspired by the LAMMPS dump file format, i.e., all relevant data is stored in one base directory which contains (i) a file named  and (ii) multiple files named . The former is of the form

and contains all information about the simulation box and the underlying discrete grid. The values in the  category define the borders of the simulation box, which is assumed to be rectangular, the  category contains the shape of the grid, and  contains the coordinates of all grid points. If the data is based on an evenly spaced rectangular grid and the grid points are	given in rising order, the  category tells  in what kind of	multidimensional array the data should be cast for representation. The files named  contain all data that varies in time. The index should rise in time and could be chosen as the number of timesteps for example. The data files should be of the following form:

Here, the names describe the scalar field written to the column beneath, e.g., density. The columns are then filled by the values of the respective field at each grid point in the same order as in . The columns can be separated by spaces, tabs, commas, semicolons, colons, or a vertical bar. The  category contains the number of timesteps corresponding to the contained data and the  category the corresponding (physical) time. If you have data of this type in the directory , it can be loaded with  via

field_trajectory = amep.load.traj(
    "/path/to/data",
    mode = "field",
    delimiter = " ",
    dumps = "field_*.txt",
    reload = True
)

The mode keyword tells  to expect field data, delimiter specifies the delimiter used for the columns in the data files, dumps takes a regular expression that matches the name format of the variable data files, and the keyword reload tells  whether to look for an existing  file and use it if it exists or to create a new one from the raw data. This should be used when your base data has changed between analysis runs. An exemplary dataset can be downloaded from <https://github.com/amepproject/amep/tree/main/examples>.










§ CONCLUSION


 is a powerful Python library for analyzing simulation data of active matter systems. It provides a unified framework for handling both particle-based and continuum simulation data and combines it with an easy-to-learn Python API. With , one can quickly analyze and plot simulation results and develop new analysis methods utilizing 's features and its seamless integrability to powerful scientific Python libraries through NumPy arrays. While its collection of analysis methods is primarily targeted at the active matter community, 's general design allows to apply  to almost any particle-based or continuum simulation data as obtained from classical molecular-dynamics and Brownian-dynamics simulations, or any kind of numerical solutions of partial differential equations.

We have exemplarily shown the potential of  by analyzing particle-based systems of more than 10^6 particles and continuum simulations with up to 10^5 grid points as typically used in active matter research.  enables the analysis of a broader class of simulation data compared to most other analysis libraries. Such simulation data comprises “dry” particle-based models such as the active Brownian particle model and its various relatives, “wet” models that explicitly model surrounding fluids including particles that are coupled to flow fields, and continuum models such as the active model B+, the Keller-Segel model, or the Cahn-Hilliard model. Applied to such models,  can provide essential insights, e.g., into phase separation, pattern formation, and critical phenomena in active matter systems. In addition to the presented observables, many more observables will be included in the future, for instance, entropy production using the method introduced in Ref. <cit.>, finite-size scaling analysis, and cluster tracking. We would like to encourage the active matter community to contribute to this open-source project in order to complement it with future analysis methods for particle-based and continuum simulation data.








§ CREDIT AUTHORSHIP CONTRIBUTION STATEMENT

Lukas Hecht: conceptualization, data curation, formal analysis, funding acquisition, investigation, methodology, project administration, software, validation, visualization, writing (original draft and review & editing). Kay-Robert Dormann: formal analysis, methodology, software, visualization, writing (original draft and review & editing). Kai Luca Spanheimer: formal analysis, methodology, software, visualization, writing (original draft and review & editing). Mahdieh Ebrahimi: formal analysis, software, visualization, writing (original draft and review & editing). Malte Cordts: software. Suvendu Mandal: formal analysis, investigation, software, visualization, writing (original draft and review & editing). Aritra K. Mukhopadhyay: formal analysis, methodology, software, visualization, writing (original draft and review & editing). Benno Liebchen: conceptualization, funding acquisition, project administration, supervision, writing (review & editing).




§ DECLARATION OF COMPETING INTEREST

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.




§ ACKNOWLEDGEMENTS

L.H. and B.L. gratefully acknowledge the computing time provided to them at the NHR Center NHR4CES at TU Darmstadt (project number p0020259). This is funded by the Federal Ministry of Education and Research, and the state governments participating on the basis of the resolutions of the GWK for national high performance computing at universities (<www.nhr-verein.de/unsere-partner>). L.H. gratefully acknowledges the support by the German Academic Scholarship Foundation (Studienstiftung des deutschen Volkes). A.K.M. and B.L. acknowledge financial support from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through project number 233630050 (TRR-146).











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