# Leveraging Large Language Models for Enhanced Product Descriptions in
eCommerce

Jianghong Zhou  Bo Liu  Jhalak Nilesh Acharya
Yao Hong and Kuang-chih Lee and Musen Wen
Walmart Global Tech, Sunnyvale, CA, USA
{jianghong.zhou, bo.liu1<EMAIL_ADDRESS>
{hong.yao0, kuangchih.lee<EMAIL_ADDRESS>
###### Abstract

In the dynamic field of eCommerce, the quality and comprehensiveness of
product descriptions are pivotal for enhancing search visibility and customer
engagement. Effective product descriptions can address the ’cold start’
problem, align with market trends, and ultimately lead to increased click-
through rates. Traditional methods for crafting these descriptions often
involve significant human effort and may lack both consistency and
scalability. This paper introduces a novel methodology for automating product
description generation using the LLAMA 2.0 7B language model. We train the
model on a dataset of authentic product descriptions from Walmart, one of the
largest eCommerce platforms. The model is then fine-tuned for domain-specific
language features and eCommerce nuances to enhance its utility in sales and
user engagement. We employ multiple evaluation metrics—including NDCG,
customer click-through rates, and human assessments—to validate the
effectiveness of our approach. Our findings reveal that the system is not only
scalable but also significantly reduces the human workload involved in
creating product descriptions. This study underscores the considerable
potential of large language models like LLAMA 2.0 7B in automating and
optimizing various facets of eCommerce platforms, offering significant
business impact, including improved search functionality and increased sales.

##  1 Introduction

The advent of eCommerce has revolutionized the way consumers engage with
products, making online visibility and customer interaction crucial aspects
for business success. A central element to this online interaction is the
product description, which significantly influences search visibility and
customer engagement Bijmolt et al. (2018). Historically, the creation of
effective product descriptions has been a manual, labor-intensive process with
a tendency to lack both consistency and scalability Zhu et al. (2019).

Moreover, novel products often face the ’cold start’ problem, where they lack
sufficient engagement data to be adequately featured or recommended by
eCommerce platforms Wang et al. (2020). Effective product descriptions have
the potential to mitigate this issue by aligning with current market trends,
thereby enhancing click-through rates Cakmak et al. (2019).

To address the existing challenges in eCommerce, this paper introduces an
innovative methodology that employs the LLAMA 2.0 7B language model to
automate the generation of product descriptions Touvron et al. (2023). We
begin by training the model on a carefully curated dataset of authentic
product descriptions from Walmart, a global leader in the eCommerce arena Zhou
and Agichtein (2020). During the initial training phase, we identify items
with high recent click-through rates and use their product descriptions as
positive training samples. Conversely, items with lower engagement rates are
used as negative training samples. For the fine-tuning process, we focus on
five specific aspects of the product description: language appeal, factual
information, product dimensions, unique attributes, and brand-related
guarantees. The fine-tuned model aims to incorporate language that captures
consumer interest while providing essential information for informed product
selection Zhou et al. (2020). This nuanced approach significantly enhances the
model’s ability to boost both sales and customer engagement Bijmolt et al.
(2018). In the second phase of our methodology, we target items that have
lackluster product descriptions for enrichment. Utilizing the fine-tuned
model, we augment these descriptions by emphasizing the aforementioned key
aspects. We validate the effectiveness of our approach using a comprehensive
set of evaluation metrics, including Normalized Discounted Cumulative Gain
(NDCG), customer click-through rates, and human evaluations. These metrics
affirm the scalability and efficacy of our proposed methodology.

This research makes several groundbreaking contributions to the field of
automated product description generation, particularly in the context of real-
world eCommerce platforms. These are as follows:

  1. 1.

First Application of LLMs: We are the first to apply Large Language Models
(LLMs), specifically LLAMA 2.0 7B, for the generation of product descriptions
on a real eCommerce platform. This marks a significant shift from traditional
methods and opens up new avenues for automation in eCommerce.

  2. 2.

Evaluation Metrics: Our research introduces a set of new and concrete
evaluation methods designed to measure the aspects of generated content that
are most pertinent to both sellers and consumers. This approach allows for a
more nuanced understanding of the model’s performance in real-world scenarios.

  3. 3.

Business and Industry Impact: The methodology and technologies developed in
this research have far-reaching implications for the eCommerce industry. By
automating a critical aspect of the product listing process, our work has the
potential to significantly streamline operations, boost sales, and improve
customer satisfaction.

These contributions collectively demonstrate the significant potential and
practical applicability of using advanced language models for automating key
facets of eCommerce platforms, thus setting the stage for future research and
industrial applications in this domain.

The remainder of this paper is organized as follows: Section 2 reviews related
work, Section 3 discusses the methodology, Section 4 presents experimental
results, and Section 6 concludes the paper and outlines future work
directions.

##  2 Related Work

Natural Language Processing (NLP) has seen substantial advancements in recent
years, thanks partly to the development of Large Language Models (LLMs). These
models have applications in various domains, from machine translation to
sentiment analysis Brown et al. (2020); Zhou et al. (2017); Lin et al. .
However, our work uniquely contributes to this landscape by focusing on the
specific use case of automated product description generation for e-Commerce
platforms.

###  2.1 Large Language Models in NLP

Large Language Models (LLMs) such as GPT-2, GPT-3, and BERT have set new
standards across a variety of NLP benchmarks, owing largely to their
capability to generate fluent and human-like text Radford et al. (2019); Brown
et al. (2020). Beyond benchmarks, these advanced models have proven utility in
practical applications including automated customer service, conversational
agents, and text summarization Adiwardana et al. (2020); Lewis et al. (2020).
LLAMA, a newly introduced open-source LLM from Meta AI, offers enhanced
scalability and fine-tuning capabilities compared to previous models Anon
(2022). In particular, the 7B-parameter version achieves state-of-the-art
performance among open-source foundation models of similar scale. This
relatively efficient model size makes LLAMA-7B well-suited for further
exploration and downstream tasks. Our work represents the first initiative to
fine-tune and apply LLAMA-7B for automated generation of engaging, high-
quality product descriptions in the eCommerce domain.

###  2.2 NLP in e-Commerce

NLP techniques have been widely applied in e-Commerce for various tasks
including sentiment analysis, recommendation systems, search engine
optimization, and more Aksnes (2019); Kumar et al. (2018). However, the
generation of engaging product descriptions remains largely a manual task
requiring significant human effort.

Prior works have explored using NLP for product attribute extraction Van-Tu
and Anh-Cuong (2016), generating stylistic variations of descriptions Chen et
al. (2019), and producing multilingual descriptions Kuznetsov and Gurevych
(2020). While promising, these approaches have fallen short of generating
high-quality, human-written product descriptions at scale.

The application of NLP in business contexts is not new, but measurable impact
in terms of revenue and customer engagement has been less explored Kumar et
al. (2018). Our work helps fill this gap by quantifying the business and
industry impact of automated product description generation using concrete
metrics like click-through rate, conversion rate, and sales.

Overall, our approach represents the first solution to effectively apply
state-of-the-art NLP techniques to automate the creation of tailored, engaging
product descriptions in e-Commerce. The scalability and business value of this
approach are demonstrated through extensive experiments.

##  3 Methodology

Our methodology employs a specialized, multi-faceted approach for the
automated generation of product descriptions, specifically targeting five key
aspects: language appeal, factual information, product dimensions, unique
attributes, and brand-related guarantees. The methodology is implemented in
three main phases: Aspect-based Segmentation, Aspect-oriented Fine-Tuning, and
Description Assembly & Evaluation.

Figure 1: Workflow of the methodology for automating product description
generation using the LLAMA 2.0 7B language model.

###  3.1 Aspect-based Segmentation

The first phase involves dividing each product description into its
constituent aspects: language appeal, factual information, product dimensions,
unique attributes, and brand-related guarantees. Custom prompts are designed
to query these specific types of information from the primary dataset, which
is sourced from Walmart’s comprehensive product catalogue. This approach
allows for targeted improvements during the subsequent fine-tuning phase.

Table 1: Prompts for Extracting Aspects of Product Descriptions Aspect | Prompt
---|---
Language Appeal | Extract the most appealing phrases from this description.
Factual Information | Identify the features and specifications from this description.
Product Dimensions | Extract dimensions and weight from this description.
Unique Attributes | Identify unique attributes from this description.
Brand-Related Guarantees | Extract any brand guarantees or warranties from this description.

###  3.2 Aspect-oriented Fine-Tuning

After the segmentation, we fine-tune the LLAMA 2.0 7B model on each of these
aspects individually, using the associated click-through rates (CTR) as
guiding metrics. The fine-tuning process incorporates an objective function
that combines the language model likelihood with the aspect-specific CTRs.
This dual objective ensures that the model produces text that is not only
linguistically coherent but also tailored to maximize consumer engagement and
clicks.

The objective of our methodology is to fine-tune a large language model for
generating product descriptions that enhance both user engagement and click-
through rates. The model fine-tuning consists of two major components:
language model likelihood and CTR optimization.

####  3.2.1 Objective Function

Our task involves optimizing a composite objective function to train the
model, as given below:

$\mathcal{L}(\theta)=\lambda\mathcal{L}_{\text{NLL}}(\theta)+(1-\lambda)\mathcal{L}_{\text{CTR}}(\theta)$
(1)

Here:

  * •

$\mathcal{L}_{\text{NLL}}(\theta)$: Represents the Negative Log-Likelihood,
aimed at generating text that is linguistically coherent.

  * •

$\mathcal{L}_{\text{CTR}}(\theta)$: This is the CTR-oriented loss function
aimed at generating text that is likely to be clicked.

  * •

$\lambda$: A hyperparameter to balance the two components of the objective
function.

The choice of $\lambda$ impacts how much weight is given to each component,
thereby allowing us to tailor the model for different business needs.

####  3.2.2 CTR Modeling

For the CTR-based component of our model, we employ logistic regression as a
simplistic yet effective approach. For each generated product description $d$,
the CTR $y_{d}$ can be modeled as:

$y_{d}=\sigma(\mathbf{w}^{T}\mathbf{x}_{d}+b)$ (2)

Here:

  * •

$\sigma$ represents the logistic sigmoid function, which transforms the model
output into a probability.

  * •

$\mathbf{x}_{d}$ is a feature vector that contains attributes of the
description $d$.

  * •

$\mathbf{w}$ and $b$ are the learned weights and bias, respectively.

The loss function $\mathcal{L}_{\text{CTR}}(\theta)$ is the Negative Log
Likelihood of the observed clicks:

$\mathcal{L}_{\text{CTR}}(\theta)=-\sum_{d}[y_{d}\log(\hat{y}_{d})+(1-y_{d})\log(1-\hat{y}_{d})]$
(3)

where $\hat{y}_{d}$ is the predicted CTR.

####  3.2.3 Negative Log-Likelihood (NLL)

The Negative Log-Likelihood loss, denoted as
$\mathcal{L}_{\text{NLL}}(\theta)$, aims to optimize the language model for
generating text sequences $s=[w_{1},w_{2},\ldots,w_{n}]$. Mathematically, it
is defined as:

$\mathcal{L}_{\text{NLL}}(\theta)=-\sum_{i=1}^{n}\log P(w_{i}|w_{<i};\theta)$
(4)

where $P(w_{i}|w_{<i};\theta)$ represents the conditional probability of
generating the $i$-th word $w_{i}$ given its preceding sequence
$w_{<i}=[w_{1},\ldots,w_{i-1}]$ according to the model’s parameters $\theta$.

The loss is computed by forward-propagating each input sequence through the
model to obtain the output probability distribution, and then using
categorical cross-entropy as a specific form of NLL to compute the loss
between the output and target sequences. The objective is to minimize this
loss to train a model that can generate high-likelihood text sequences.

###  3.3 Description Assembly and Evaluation

In the evaluation phase, the model is prompted to generate content for each of
the five specified aspects. The generated content for each aspect is then
assembled to construct a complete, coherent product description. We employ a
series of evaluation metrics, including Normalized Discounted Cumulative Gain
(NDCG), customer click-through rates, and human assessments, to validate the
effectiveness of our methodology.

##  4 Experiments

###  4.1 Dataset and Preprocessing

For our experiments, we utilize the Walmart relevance items dataset, a
comprehensive collection of product descriptions and their associated
relevance metrics. This dataset is pivotal for our analysis as it provides a
real-world representation of products on one of the world’s largest e-commerce
platforms. To ensure robustness and accuracy, we divide the dataset into two
main subsets:

  1. 1.

Training Subset: This consists of the top 50% of items from the dataset,
categorized based on their relevance. These items are deemed as high-quality
samples and are employed to train and fine-tune our LLAMA 2.0 7B model.

  2. 2.

Testing Subset: The lower 50% of items, which might not be optimally
described, form this subset. We aim to evaluate the performance of our trained
model on these items to ascertain its effectiveness in real-world scenarios.

###  4.2 Model Training and Fine-tuning

With the training subset in place, we embark on the task of training the LLAMA
2.0 7B model. Leveraging the inherent prowess of LLAMA in understanding and
generating text, we believe that fine-tuning it on our dataset will endow it
with the ability to generate product descriptions that resonate with
e-commerce consumers.

###  4.3 Evaluation Metrics

To ensure a comprehensive and robust evaluation of our model’s performance, we
adopt a combination of automated and human-centric metrics:

  * •

BM25: An esteemed ranking function in the field of information retrieval, BM25
assesses the semantic relevance of the generated product descriptions. By
gauging how closely the model-generated descriptions align with optimal
product descriptions, we aim to obtain a measure of the quality and relevance
of our model’s outputs.

  * •

Human-Evaluation-based NDCG@10: Recognizing the importance of human perception
in the context of product descriptions, we also integrate a human-centric
evaluation metric. We recruit volunteers to rate the generated descriptions on
a scale of 1 to 5. These scores are then employed to compute the Normalized
Discounted Cumulative Gain (NDCG), a standard metric that measures the ranking
quality. This approach provides insights into the practical utility and appeal
of the descriptions generated by our model from an end-user perspective.

Through the amalgamation of BM25 and NDCG, our evaluation strategy aims to
offer both objective and subjective perspectives on the model’s efficacy,
ensuring a holistic assessment of its capabilities in the e-commerce domain.

Table 2: Comparison of Original and Enhanced Product Description Aspects Aspect |  Original Description |  Enhanced Description
---|---|---
Appealing Introduction |  Get in on a cool-casual style with Terra & Sky’s Jeggings for Women. |  Dive into an effortlessly chic style with Terra & Sky’s exclusive Women’s Jeggings, tailored just for you.
Factual Information |  Material: 61% Cotton/24% Polyester/14% Rayon/1% Spandex. Care: Machine washable. Country of Origin: Imported. |  Crafted with a premium blend of 61% Cotton, 24% Polyester, 14% Rayon, and 1% Spandex, these jeggings assure durability and longevity.
Product Dimensions |  Size: Model is 5’11” and is wearing a size 1X. Fit: Skinny fit. Rise and Inseam: High rise; 28” inseam. |  Specifically designed for a flattering silhouette, these jeggings come in a high-rise style with a 28” inseam.
Unique Attributes |  The inner elasticized waist and stretch denim fabric provide a comfortable fit. |  Stand out with the jeggings’ inner elasticized waist and stretch denim fabric.
Brand-Related Guarantees |  Only at Walmart. |  Terra & Sky redefines elegance, exclusive at Walmart.
Pairing Tip |  Pair these with your favorite graphic tee. |  Team up these jeggings with a chic top.
Series |  Women’s Plus Size Jeans from Terra & Sky |  Part of the Women’s Plus Size Jeans collection by Terra & Sky.
Figure 2: Variation of BM25 score with $\lambda$. The peak performance is
observed at $\lambda=0.429$.

###  4.4 Results and Discussion

In this section, we present and discuss the results of our experiments.

The experimental outcomes offer substantial insights into the capabilities of
our approach, especially when enhancing product descriptions using the LLAMA
2.0 7B model. Figures 3 and 4 serve as pivotal points for our discussion.

Starting with the BM25 scores, a marked improvement from 66.44 (bottom 50%) to
78.65 (enhanced) showcases the model’s capacity for semantic alignment with
high-quality descriptions. While there remains a slight gap compared to the
top 50% score of 82.76, the difference is narrowing, hinting at the promise of
our methodology.

Human-evaluated NDCG scores further fortify our findings. The enhancement from
an NDCG score of 0.68 to 0.76 illustrates that our model-generated
descriptions resonate well with human evaluators, inching closer to the top-
tier score of 0.82. This underscores the holistic improvements our methodology
brings, both in clarity and appeal.

Several implications emerge:

  * •

The pivotal role of fine-tuning is evident, emphasizing its significance in
tailored tasks.

  * •

A discernible gap between enhanced and top-tier scores signals opportunities
for further refinement.

  * •

The tested methodology, while applied on Walmart’s dataset, suggests broader
e-commerce applicability.

###  4.5 Case Study

The enhancement of product descriptions is vital for e-commerce platforms,
especially when it can lead to improved customer engagement and increased
sales. Our methodology demonstrates practicality and effectiveness, as
observed in the transformation of a sample product description from Walmart.

###  4.6 Description Context

The product under consideration is Terra & Sky’s Jeggings for Women. As one of
Walmart’s apparel offerings, it represents a standard product category with
myriad similar listings. The challenge lies in making the product stand out
and appeal more to potential buyers.

###  4.7 Enhancement Overview

Our methodology aims to improve various aspects of product descriptions. The
results are detailed in Table 2, which presents a side-by-side comparison of
the original and enhanced descriptions. As evident, the new descriptions are
not only more concise but also capture the essence of the product more
effectively.

###  4.8 Practical Implications

Several key takeaways from the case study include:

  * •

Appeal Enhancement: The enhanced description positions the product more
attractively, making it more likely for potential buyers to consider
purchasing.

  * •

Clarity: By focusing on distinct aspects and presenting them clearly,
potential buyers can quickly grasp the essential features of the product,
reducing decision-making time.

  * •

Branding: The refined description emphasizes brand exclusivity, potentially
enhancing brand value and trustworthiness in the eyes of the customer.

This case study affirms the practical effectiveness of our approach. By
employing our methodology, e-commerce platforms can enhance product listings
en masse, improving overall platform attractiveness and customer engagement.

In summation, our results solidify the potential of integrating large language
models in e-commerce. As AI-driven techniques become more refined, it is
conceivable to anticipate a deep synergy between e-commerce and sophisticated
models in the near future.

Figure 3: Comparative results of BM25 scores. Figure 4: Comparative results of
Human-Evaluation-based NDCG@10 scores.

##  5 Ablation Study

In our endeavor to understand the impact of the hyperparameter $\lambda$ on
our model’s performance, we conducted an ablation study. The parameter
$\lambda$ plays a pivotal role in modulating the trade-off between the model’s
objectives, which has significant implications for its efficacy in generating
relevant product descriptions.

Referring to Figure 2, it is evident that the BM25 score exhibits an optimal
value at $\lambda=0.429$. Intuitively, this demonstrates that a careful
balance between our model’s objectives, modulated by $\lambda$, is crucial for
achieving the best results. Beyond this point, it’s possible that the model
over-prioritizes one objective over the other, leading to sub-optimal
performance. The noise in the graph and the shaded region representing one
standard deviation provide insights into the inherent variability of real-
world data and underline the robustness of our results Zhou et al. (2021).

###  5.1 Discussion

The ablation study’s findings underscore the significance of hyperparameter
tuning. It emphasizes that even in sophisticated models driven by large
amounts of data, nuanced adjustments to hyperparameters can have pronounced
effects on performance. This investigation into the behavior of $\lambda$ not
only informs our understanding but also paves the way for future work, where
adaptive techniques might be employed to optimize such parameters dynamically.

##  6 Conclusion

In this work, we have investigated the potential of state-of-the-art language
models, with a particular focus on the LLAMA 2.0 7B, for the purpose of
enhancing product descriptions in e-commerce platforms. Our methodology
incorporated a dataset from Walmart, and we employed a differentiated strategy
for model training using both high and low engagement product descriptions.

The framework we introduced prioritizes five essential aspects of product
descriptions, facilitating a more structured and targeted approach to
description enhancement. Through empirical evaluations, it was observed that
the BM25 and NDCG scores for descriptions improved post-enhancement,
indicating the potential of our model in terms of improving semantic relevance
and overall user engagement.

Furthermore, our ablation study on the hyperparameter $\lambda$ has provided
an understanding of its influence on the BM25 scores, showcasing the
importance of fine-tuning model parameters to achieve optimal performance. The
nuanced observations from this study are significant for researchers aiming to
optimize language models for similar tasks.

In summation, this research contributes to the growing body of knowledge
surrounding the application of large language models in practical e-commerce
scenarios. While the results presented are promising, they also pave the way
for further investigations, especially in the realm of NLP-driven automated
content generation.

##  7 Limitations

Our methodology has shown promising results in leveraging LLAMA 2.0 7B for
enhancing product descriptions in the e-commerce domain. While our approach
offers substantial improvements, there are aspects worth considering for
future refinements:

  1. 1.

Adaptability Across Platforms: The study’s foundation is based on data from
Walmart, one of the global leaders in e-commerce. Although this provides a
robust baseline, it would be valuable to test the adaptability of our model
across different e-commerce platforms, offering an even broader perspective.

  2. 2.

Tuning Parameters: The optimal value of $\lambda$ in our study offers an
excellent starting point for fine-tuning, but further research can explore its
sensitivity across different product categories or datasets to optimize
results even more.

  3. 3.

Universal Applicability: Every language model, including LLAMA 2.0 7B, learns
from its data, reflecting the diversity and depth of its training material.
Future iterations might focus on ensuring even broader representation in the
enhanced descriptions, making them universally appealing.

  4. 4.

Efficiency Optimizations: Our approach is inherently scalable, yet as with any
advanced system, there are always avenues to further enhance computational
efficiency, especially for real-time processing.

  5. 5.

Refining Evaluation Metrics: The human-based NDCG evaluations provided
significant insights into the efficacy of our approach. Exploring additional
evaluation metrics might offer even more nuanced understandings of user
preferences and needs.

We view these areas not as shortcomings, but as opportunities for further
refinement and exploration in the ever-evolving domain of automated content
generation. This study serves as a stepping stone, and we are optimistic about
the advancements that future research will bring to this field.

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# Exact moments and re-entrant transitions in the inertial dynamics of active
Brownian particles

Manish Patel  Institute of Physics, Sachivalaya Marg, Bhubaneswar 751005,
India  Homi Bhabha National Institute, Anushaktinagar, Mumbai 400094, India
<EMAIL_ADDRESS>Debasish Chaudhuri  Institute of Physics, Sachivalaya
Marg, Bhubaneswar 751005, India  Homi Bhabha National Institute,
Anushaktinagar, Mumbai 400094, India<EMAIL_ADDRESS>
###### Abstract

In this study, we investigate the behavior of free inertial Active Brownian
Particles (ABP) in the presence of thermal noise. While finding a closed-form
solution for the joint distribution of positions, orientations, and velocities
using the Fokker-Planck equation is generally challenging, we utilize a
Laplace transform method to obtain the exact temporal evolution of all
dynamical moments in arbitrary dimensions. Our expressions in $d$ dimensions
reveal that inertia significantly impacts steady-state kinetic temperature and
swim pressure while leaving the late-time diffusivity unchanged. Notably, the
velocity distribution exhibits a remarkable re-entrant transition from passive
Gaussian to active non-Gaussian behaviors. We construct a phase diagram using
the exact expression of the $d$-dimensional kurtosis as a function of activity
and inertia. Our analytic expressions describe steady states and offer
insights into time-dependent crossovers observed in various velocity and
displacement moments.

##  1 Introduction:

Active elements can self-propel by consuming and dissipating ambient or
internal energy locally, breaking detailed balance and the equilibrium
fluctuation-dissipation relation [1, 2, 3, 4]. Natural examples of active
agents are found across multiple scales, from motor proteins, cells, and
tissues to insects, birds, fishes, and animals [5, 6, 7, 8, 9, 10, 11, 12].
Although relatively less, a number of artificial active systems of micro to
macro scales have been engineered over the years [1, 2, 13, 14, 15]. Active
agents performing persistent random motion are described as active Brownian
particles (ABP). Their dynamics up to the second moment agree with other
related models, e.g., the run-and-tumble particles (RTP) and active Ornstein-
Uhlenbech (AOUP) process [16, 17, 18, 19]. They are often assumed to perform
overdamped dynamics, as, e.g., the inertial relaxation time of active colloids
$\sim 100$ ns is negligibly small with respect to the persistence time.
However, inertial relaxation can be much slower for larger active elements,
including insects to animals and macro-sized artificial active matter, and as
a result, can significantly influence the resultant dynamics [10, 11, 12, 13,
20, 14, 15, 21, 22, 23, 24, 25, 26, 27]. For example, inertia can lead to the
disappearance of motility-induced phase separation [28, 29, 30] observed for
over-damped ABPs [31, 32, 33, 34] and taming down of the instability in active
nematics [35]. Recent experiments on vibrobots showed an inertial delay
between the instantaneous orientations and velocities and bimodality in the
velocity distributions [14, 29] captured by perturbative calculations [23].
Nevertheless, even with substantial advancements in active matter research, a
thorough understanding of the dynamics of non-interacting inertial ABPs still
remains elusive.

In this paper, we consider motion of non-interacting ABPs in the presence of
translational inertia. Using a Laplace transform approach, we show that the
Fokker-Planck equation governing the dynamics can be utilized to evaluate the
exact time dependence of all moments of dynamical variables in arbitrary
$d$-dimensions. This is the first main contribution of this paper. Notably,
the analytic method was originally proposed back in 1952 to study equilibrium
properties of worm-like chains [36] and was recently put in use to study the
dynamics of overdamped ABPs in Ref. [19, 37, 38, 39].

We present exact expressions of several dynamical moments, which agree with
the direct numerical integration results. The system eventually reaches a
steady state in velocity but not displacement, with mean-squared displacement
undergoing dynamic crossovers before reaching asymptotic diffusion with
inertia-independent diffusivity. In contrast, the kinetic temperature and swim
pressure depend on inertia; with inertia, while the kinetic temperature
increases to saturate eventually, the swim pressure decreases monotonically to
vanish in the infinite mass limit.

The most striking result is the occurrence of a re-entrant transition between
active and passive states as inertia varies, quantified through deviations
from the equilibrium-like Gaussian velocity distribution. Analytically, this
is captured by the $d$-dimensional kurtosis of velocity, which shows a non-
monotonic variation with inertia and a decrease to negative values followed by
a saturation with increasing activity. We use the results to obtain a phase
diagram in the activity-inertia plane. This is the second main contribution of
this study. Note that in this single-particle system, there is no genuine
phase transition; instead, it represents a crossover between active and
passive behaviors. Further, we propose approximate analytic forms of
probability distributions of velocity that show reasonable agreement with
numerical results at small and large inertia limits and can qualitatively
capture the re-entrant active-passive crossovers.

The paper is structured as follows: In Section 2, we introduce the model and
Langevin dynamics. The subsequent section outlines a Laplace-transform-based
approach to compute dynamic moments in any dimension. Sections 4 and 5 present
detailed results related to velocity and displacement moments. Section 4
delves into discussions about kinetic temperature, the phase diagram, and
velocity distributions, while Section 5 explores topics like active
diffusivity and swim pressure. Finally, in Section 6, we provide a conclusion,
summarizing our key findings.

##  2 Model

The underdamped motion of ABPs in $d$-dimensional space moving with a constant
active speed $\lambda^{\prime}$ in the orientation
$\hat{\bm{u}}=(u_{1},u_{2},...,u_{d})$ is described by its position
${\mathbf{r}}^{\prime}=(r_{1}^{\prime},r_{2}^{\prime},...,r_{d}^{\prime})$ and
velocity
${\mathbf{v}}^{\prime}=(v_{1}^{\prime},v_{2}^{\prime},...,v_{d}^{\prime})$
evolving with time $t^{\prime}$. We consider the dynamics in the presence of
translational and orientational Brownian noise with diffusivities $D$ and
$D_{r}$, respectively. The time and length scales are set by
$\tau_{r}=D_{r}^{-1}$ and $\ell=\sqrt{D/D_{r}}$, to give the unit of velocity
$\bar{v}=\sqrt{DD_{r}}$. We use dimensionless position
${\mathbf{r}}={\mathbf{r}}^{\prime}/\ell$, time $t=t^{\prime}/\tau_{r}$, and
velocity ${\mathbf{v}}={\mathbf{v}}^{\prime}/\bar{v}$ to express the Langevin
equation of motion in $d$ dimensions in the following Ito form [40, 41, 42]

$\displaystyle\mathrm{d}r_{i}$ $\displaystyle=$ $\displaystyle
v_{i}\mathrm{d}t,$ (1) $\displaystyle M\mathrm{d}v_{i}$ $\displaystyle=$
$\displaystyle-(v_{i}-\lambda u_{i})\mathrm{d}t+\sqrt{2}\,\mathrm{d}B_{i}(t),$
(2) $\displaystyle\mathrm{d}u_{i}$ $\displaystyle=$
$\displaystyle(\delta_{ij}-u_{i}u_{j})\,\sqrt{2}\,\mathrm{d}B_{j}(t)-(d-1)u_{i}\mathrm{d}t.$
(3)

where we used the dimensionless mass $M=\tau/\tau_{r}$ with inertial
relaxation time $\tau=m/\gamma$, $m$ denoting the translational inertia and
$\gamma$ the viscous damping, and Péclet number
$\lambda=\lambda^{\prime}/\sqrt{DD_{r}}$. The Gaussian noise in translation
and rotation are uncorrelated, and their components obey
$\langle\mathrm{d}B_{i}\rangle=0$ and
$\langle\mathrm{d}B_{i}\mathrm{d}B_{j}\rangle=\delta_{ij}\,\mathrm{d}t$. The
first term in Eq. (3) denotes a projection operator for the noise onto a
$(d-1)$-dimensional surface perpendicular to $\hat{\bm{u}}$. The second term
ensures that the unit vector remains normalized after displacement. The
equations can be directly integrated numerically using the Euler-Maruyama
scheme.

##  3 Calculation of moments from the Fokker-Planck equation

Noting that the heading direction $\hat{\bm{u}}$ undergoes an orientational
diffusion on a $(d-1)$-dimensional hypersurface and ${\mathbf{v}}$ performs a
drift-diffusion, the probability distribution
$P({\mathbf{r}},{\mathbf{v}},\hat{\bm{u}},t)$ evolves following the Fokker-
Planck equation

$\displaystyle\partial_{t}P=-\nabla\cdot({\mathbf{v}}\,P)-(1/M)\nabla_{v}\cdot[(\lambda\hat{\bm{u}}-{\mathbf{v}})P]+(1/M^{2})\nabla_{v}^{2}P+\nabla_{u}^{2}P$
(4)

where $\nabla$ and $\nabla_{v}$ denote the gradient operator in
$d$-dimensional position and velocity space, respectively. The Laplacian
$\nabla_{u}^{2}$ in $(d-1)$-dimensional orientation space can be expressed in
terms of Cartesian coordinates ${\bf y}$ as
$\nabla_{u}^{2}=y^{2}\sum_{i=1}^{d}\partial_{yi}^{2}-[y^{2}\partial_{y}^{2}+(d-1)y\partial_{y}]$
by defining $u_{i}=y_{i}/y$ with $y=|\bf{y}|$. Using a Laplace transform
$\tilde{P}({\mathbf{r}},{\mathbf{v}},\hat{\bm{u}},s)=\int_{0}^{\infty}dt\,e^{-st}\,P({\mathbf{r}},{\mathbf{v}},\hat{\bm{u}},t)$,
the Fokker-Planck equation can be expressed as

$\displaystyle-P({\mathbf{r}},{\mathbf{v}},\hat{\bm{u}},0)+s\tilde{P}({\mathbf{r}},{\mathbf{v}},\hat{\bm{u}},s)=-\nabla\cdot({\mathbf{v}}\tilde{P})-\frac{1}{M}\nabla_{v}\cdot[(\lambda\hat{\bm{u}}-{\mathbf{v}})\tilde{P}]+\frac{1}{M^{2}}\nabla_{v}^{2}\tilde{P}+\nabla_{u}^{2}\tilde{P}$

Therefore, the mean of any observable $\langle\psi\rangle_{s}=\int
d{\mathbf{r}}\,d{\mathbf{v}}\,d\hat{\bm{u}}\,\psi({\mathbf{r}},{\mathbf{v}},\hat{\bm{u}})\tilde{P}({\mathbf{r}},{\mathbf{v}},\hat{\bm{u}},s)$
satisfies the moment equation

$\displaystyle-\langle\psi\rangle_{0}+s\langle\psi\rangle_{s}$
$\displaystyle=\langle{\mathbf{v}}\cdot\nabla\psi\rangle_{s}+\frac{\lambda}{M}\langle\hat{\bm{u}}\cdot\nabla_{v}\psi\rangle_{s}-\frac{1}{M}\langle{\mathbf{v}}\cdot\nabla_{v}\psi\rangle_{s}+\frac{1}{M^{2}}\langle\nabla_{v}^{2}\psi\rangle_{s}+\langle\nabla_{u}^{2}\psi\rangle_{s},$
(5)

where $\langle\psi\rangle_{0}=\int
d{\mathbf{r}}\,d{\mathbf{v}}\,d\hat{\bm{u}}\,\psi({\mathbf{r}},{\mathbf{v}},\hat{\bm{u}})P({\mathbf{r}},{\mathbf{v}},\hat{\bm{u}},0)$
is given by the initial condition, which without any loss of generality, can
be expressed as
$P({\mathbf{r}},{\mathbf{v}},\hat{\bm{u}},0)=\delta({\mathbf{r}})\delta({\mathbf{v}}-{\mathbf{v}}_{0})\delta(\hat{\bm{u}}-\hat{\bm{u}}_{0})$.
Equation (5) can be used to determine any dynamical moment of interest in
arbitrary d-dimensions as a function of time by performing an inverse Laplace
transform. In the following, we denote the steady-state values
$\lim_{t\to\infty}\langle\psi\rangle(t)\equiv\langle\psi\rangle_{st}=\lim_{s\to
0}s\langle\psi\rangle_{s}$. Our theoretical approach differs from the recent
works on inertial ABPs [14, 23, 43].

##  4 Velocity moments, steady states, and phase transitions

We first employ the above equation to obtain various velocity moments as they
evolve to a steady state. The dynamics of inertial passive particles lead to
the equilibrium Maxwell-Boltzmann distribution of velocity. Activity changes
this behavior [14, 29]. Here, we calculate up to fourth order cumulant of
velocity to identify such deviations and hence determine a phase diagram
describing departures from the equilibrium Gaussian characteristic.

###  4.1 Mean velocity

Using $\psi={\mathbf{v}}$ in equation (5) we get the velocity in the Laplace
space
$\langle{\mathbf{v}}\rangle_{s}=\frac{{\mathbf{v}}_{0}+\lambda\langle\hat{\bm{u}}\rangle_{s}/M}{s+1/M}$
in terms of $\langle\hat{\bm{u}}\rangle_{s}$. Again, using
$\psi=\hat{\bm{u}}$, equation (5) gives
$\langle\hat{\bm{u}}\rangle_{s}=\hat{\bm{u}}_{0}/[s+(d-1)]$. Together, these
two expressions lead to

$\langle{\mathbf{v}}\rangle_{s}=\frac{{\mathbf{v}}_{0}}{s+M^{-1}}+\frac{\lambda\hat{\bm{u}}_{0}}{M(s+d-1)(s+M^{-1})}.$
(6)

Expressions of dynamical variables with $s$ dependence
$\prod_{i}(s+s_{i})^{-1}$ lead to sum over exponentials in time after inverse
Laplace transform. Thus,
$\langle\hat{\bm{u}}\rangle(t)=\hat{\bm{u}}_{0}e^{-(d-1)t}$ and

$\displaystyle\langle{\mathbf{v}}\rangle(t)={\mathbf{v}}_{0}e^{-t/M}+\frac{\lambda\hat{\bm{u}}_{0}}{(d-1)M-1}\left(e^{-t/M}-e^{-(d-1)t}\right).$
(7)

The mean velocity vanishes in the steady state,
$\langle{\mathbf{v}}\rangle_{st}=0$.

In the presence of inertia, the orientation of the velocity vector
${\mathbf{v}}$ can be different from the heading direction $\hat{\bm{u}}$. Let
us define instantaneous velocity components parallel and perpendicular to the
heading direction,
${\mathbf{v}}_{\parallel}=({\mathbf{v}}\cdot\hat{\bm{u}})\hat{\bm{u}}$,
${\mathbf{v}}_{\perp}={\mathbf{v}}-{\mathbf{v}}_{\parallel}$. Using
$\psi=v_{\parallel}=\hat{\bm{u}}\cdot{\mathbf{v}}$ in equation(5) we get

$\displaystyle\langle
v_{\parallel}\rangle_{s}\equiv\frac{(\hat{\bm{u}}_{0}\cdot{\mathbf{v}}_{0})+\lambda/(sM)}{s+1/M+(d-1)}$
(8)

Its inverse Laplace transform yields,

$\displaystyle\langle
v_{\parallel}\rangle(t)=\frac{\lambda}{(d-1)M+1}+\left((\hat{\bm{u}}_{0}\cdot{\mathbf{v}}_{0})-\frac{\lambda}{(d-1)M+1}\right)e^{-((d-1)+M^{-1})t}$
(9)

Thus $\langle v_{\parallel}\rangle$ starts from
$\hat{\bm{u}}_{0}\cdot{\mathbf{v}}_{0}$ at $t=0$ to asymptotically reach the
steady-state value $\frac{\lambda}{(d-1)M+1}$ (see Fig.1($a$) ). In the limit
of vanishing inertia, $\langle v_{\parallel}\rangle=\lambda$, the velocity
vector always remains oriented along the heading direction. The larger the
inertia, the bigger the deviation from the heading direction (also see Fig. 5
in the Appendix). By symmetry $\langle{\mathbf{v}}_{\perp}\rangle=0$.

Figure 1: Time evolution of velocity moments $\langle v_{\parallel}\rangle$
($a$), $\langle{\mathbf{v}}^{2}\rangle$ ($c$),
$\langle{\mathbf{v}}^{4}\rangle$ ($d$) in $d=2,3$ with initial condition
${\mathbf{v}}_{0}={\bf 0}$. In ($a$), ($c$), ($d$), we use $\lambda=20$ and
$M=2$ for $d=2$ and $M=0.1$ in $d=3$. In ($b$), the delay function
${\cal{C}}({\mathbf{v}}(t),\hat{\bm{u}}(t))$ is shown at $\lambda=100$ in 2d
for two $M$ values (solid lines) and for $M=100$ in $d=3$ (dashed line). The
lines in $(a)$, $(b)$, $(c)$ and $(d)$ are the plots of equations (9), (10),
(4.3), and (A) respectively and the points denote simulation result in 2d. We
indicate crossovers between different scaling regimes in ($c$) and $(d)$. In
($c$), crossover between $\sim t$ to $\sim t^{2}$ appears at $t_{I}=0.01$.
($d$) shows crossovers from $\sim t^{2}$ to $\sim t^{3}$ at time $t_{I}=0.005$
and $\sim t^{3}$ to $\sim t^{4}$ at $t_{II}=0.04$ for $d=2$.

###  4.2 Lag function

While analyzing the motion of inertial vibrobots, a lag function was used in
Ref. [14],
${\cal{C}}({\mathbf{v}}(t),\hat{\bm{u}}(t))=\langle{\mathbf{v}}(t)\cdot\hat{\bm{u}}(0)\rangle-\langle\hat{\bm{u}}(t)\cdot{\mathbf{v}}(0)\rangle$,
which quantifies the average difference between the projection of velocity to
initial heading direction and the projection of heading direction to the
initial velocity. In equilibrium, time-reversal symmetry ensures
${\cal{C}}({\mathbf{v}}(t),\hat{\bm{u}}(t))=0$. We obtain the following
closed-from expression for inertial ABPs,

$\displaystyle{\cal{C}}({\mathbf{v}}(t),\hat{\bm{u}}(t))=\frac{\lambda}{1-[(d-1)M]^{-1}}\left(e^{-t/M}-e^{-(d-1)t}\right),$
(10)

using ${\mathbf{v}}(0)=\lambda\hat{\bm{u}}_{0}$. In the absence of inertia,
${\cal{C}}({\mathbf{v}}(t),\hat{\bm{u}}(t))=0$. In the inertial system,
${\cal{C}}({\mathbf{v}}(t),\hat{\bm{u}}(t))$ starts from zero at $t=0$ to
returns to zero as $t\to\infty$ after reaching a maximum at an intermediate
time $t_{m}=\frac{M}{M(d-1)-1}\ln[(d-1)M]$. Plots of this expression in
$d=2,3$ are shown in Fig.1($b$), with a comparison against numerical
simulations performed in two dimensions. A similar non-monotonic variation was
observed in vibrobots [14].

###  4.3 Mean squared velocity

Using $\psi={\mathbf{v}}^{2}$ in equation (5) and
$\langle\psi\rangle(0)={\mathbf{v}}_{0}^{2}$, ${\mathbf{v}}\cdot\nabla\psi=0$,
$\langle{\mathbf{v}}\cdot\nabla_{v}\psi\rangle_{s}=2\langle{\mathbf{v}}^{2}\rangle_{s}$,
$\langle\hat{\bm{u}}\cdot\nabla_{v}\psi\rangle_{s}=2\langle\hat{\bm{u}}\cdot{\mathbf{v}}\rangle_{s}$,
$\langle\nabla_{v}^{2}\rangle_{s}=2d\langle 1\rangle_{s}=2d/s$ 111As, $\langle
1\rangle_{s}=\int d{\mathbf{r}}d{\mathbf{v}}d\hat{\bm{u}}\,\tilde{P}=\int
d{\mathbf{r}}d{\mathbf{v}}d\hat{\bm{u}}\int_{0}^{\infty}dt\,e^{-st}P=\int_{0}^{\infty}dt\,e^{-st}\\{d{\mathbf{r}}d{\mathbf{v}}d\hat{\bm{u}}P\\}=\int_{0}^{\infty}dt\,e^{-st}=1/s$
and $\nabla^{2}\psi=0$, we get
$\langle{\mathbf{v}}^{2}\rangle_{s}=\frac{1}{s+2/M}\left[{\mathbf{v}}_{0}^{2}+\frac{2\lambda\langle
v_{\parallel}\rangle_{s}}{M}+\frac{2d}{sM^{2}}\right].$ Using equation(8), we
get

$\displaystyle\langle{\mathbf{v}}^{2}\rangle_{s}$
$\displaystyle=\frac{1}{s+2/M}\left[{\mathbf{v}}_{0}^{2}+\frac{2\lambda(\hat{\bm{u}}_{0}\cdot{\mathbf{v}}_{0}+\lambda/(sM))}{M(s+1/M+(d-1))}+\frac{2d}{sM^{2}}\right].$
(11)

The inverse Laplace transform gives

$\displaystyle\langle{{\mathbf{v}}}^{2}\rangle({t})=\frac{\lambda^{2}}{(d-1)M+1}+\frac{d}{M}+\left(\left({\mathbf{v}}_{0}^{2}-\frac{d}{M}\right)+\frac{2(\hat{\bm{u}}_{0}\cdot{\mathbf{v}}_{0})\lambda-\lambda^{2}}{(d-1)M-1}\right)e^{-2{t}/M}$
$\displaystyle-\frac{2\lambda[\,(\hat{\bm{u}}_{0}\cdot{\mathbf{v}}_{0})\\{(d-1)M+1\\}-\lambda\,]}{(d-1)^{2}M^{2}-1}e^{-[(d-1)+M^{-1}]{t}}$
(12)

We plot this expression in Fig. 1($c$) for $d=2,3$ and show a comparison with
numerical simulations in 2d. To analyze the observed dynamical crossovers from
$\sim t$ to $\sim t^{2}$ growth before reaching the steady state, we expand
$\langle{\mathbf{v}}^{2}\rangle(t)$ around $t=0$ using ${\mathbf{v}}_{0}={\bf
0}$,

$\displaystyle\langle{\mathbf{v}}^{2}\rangle(t)=\frac{2d}{M^{2}}t+\frac{(\lambda^{2}M-2d)}{M^{3}}t^{2}+\frac{d\left(4-\lambda^{2}M^{2}\right)+\lambda^{2}(M-3)M}{3M^{4}}t^{4}+{\cal{O}}(t^{5}).$
(13)

The expansion shows a $\langle{\mathbf{v}}^{2}\rangle\sim t$ scaling at a
short time before a crossover to $\langle{\mathbf{v}}^{2}\rangle\sim t^{2}$ at
$t_{{I}}=2dM/(\lambda^{2}M-2d)$. For crossovers starting from a different
initial condition, see C. In the asymptotic limit of long time
$t\gg\frac{M}{2}$ and $t\gg\frac{M}{M(d-1)+1}$, the mean-squared velocity
reaches the steady state value

$\displaystyle\langle{\mathbf{v}}^{2}\rangle_{st}=\frac{\lambda^{2}}{(d-1)M+1}+\frac{d}{M}.$
(14)

Given that $\langle{\mathbf{v}}\rangle_{st}=0$, the velocity fluctuation
$\langle\delta{\mathbf{v}}^{2}\rangle_{st}=\langle{\mathbf{v}}^{2}\rangle_{st}$.

###  4.4 Effective temperature and violation of equilibrium fluctuation-
dissipation relation

The expression of mean squared velocity can be used to obtain the steady-state
kinetic temperature of the system

$\displaystyle k_{B}T_{\rm
kin}=\frac{M\langle{\mathbf{v}}^{2}\rangle_{st}}{d}=1+\frac{\lambda^{2}}{d}\frac{M}{(d-1)M+1}.$
(15)

Remarkably, unlike in equilibrium, the kinetic temperature is a function of
inertia. In the over-damped limit $k_{B}T_{\rm kin}=1$. The kinetic
temperature increases with $M$ linearly at a small $M$ to saturate to
$1+\lambda^{2}/[d(d-1)]$ at large $M$.

At equilibrium ${\cal I}=D-\mu k_{B}T=0$, due to the equilibrium fluctuation-
dissipation relation. Using the expression of effective diffusivity $D_{\rm
eff}=1+\lambda^{2}/[d(d-1)]$ derived later in equation (40) in Sec. 5, we
obtain the dimensionless form of the violation of equilibrium fluctuation-
dissipation

$\displaystyle{\cal{I}}=D_{\rm eff}-k_{B}T_{\rm
kin}=\frac{\lambda^{2}}{d(d-1)}\left[1-\frac{(d-1)M}{(d-1)M+1}\right].$ (16)

Clearly, the violation is maximum in the overdamped limit of $M=0$. It
decreases linearly with $M$ at small $M$, to vanish asymptotically as
$M\to\infty$. Thus, with increasing $M$, ABPs can return to equilibrium-like
behavior.

###  4.5 Fluctuation of velocity component in the heading direction

As we have shown before,

$\displaystyle\langle v_{\parallel}\rangle_{st}=\frac{\lambda}{(d-1)M+1},$
(17)

the steady state value of $\langle v_{\parallel}\rangle_{st}$ decreases from
$\lambda$ to $0$ monotonically, with increasing mass (Fig. 6($a$) in C). Here
we calculate the variance $\langle\delta v_{\parallel}^{2}\rangle=\langle
v_{\parallel}^{2}\rangle-\langle v_{\parallel}\rangle^{2}$. Using
$\psi=v_{\parallel}^{2}\equiv(\hat{\bm{u}}\cdot{\mathbf{v}})^{2}$ in
equation(5)

$\displaystyle(s+2/M+2d)\langle
v_{\parallel}^{2}\rangle_{s}=(\hat{\bm{u}}_{0}\cdot{\mathbf{v}}_{0})^{2}+\frac{2\lambda}{M}\langle
v_{\parallel}\rangle_{s}+\frac{2}{sM^{2}}+2\langle{\mathbf{v}}^{2}\rangle_{s}$

While it is straightforward to calculate the full time-evolution of $\langle
v_{\parallel}^{2}\rangle(t)$ performing an inverse Laplace transform of the
above expression, we show the steady-state expressions

$\displaystyle\langle v_{\parallel}^{2}\rangle_{st}$ $\displaystyle=$
$\displaystyle\frac{\lambda^{2}(M+1)}{((d-1)M+1)(dM+1)}+\frac{1}{M}$ (18)
$\displaystyle\langle\delta v_{\parallel}^{2}\rangle_{st}$ $\displaystyle=$
$\displaystyle\frac{(d-1)\lambda^{2}M^{2}}{((d-1)M+1)^{2}(dM+1)}+\frac{1}{M}$
(19)

In the limit of vanishing inertia, the above expression gives $M\langle\delta
v_{\parallel}^{2}\rangle_{st}=1$ and matches with the overdamped limit of
$k_{B}T_{kin}=1$. In the other limit of $M\to\infty$, $M\langle\delta
v_{\parallel}^{2}\rangle_{st}=1+\frac{\lambda^{2}}{d(d-1)}$, again agrees with
the $M\to\infty$ limit of $k_{B}T_{kin}$. It is important to note that, in
general, for intermediate values of $M$, $M\langle\delta
v_{\parallel}^{2}\rangle_{st}\neq k_{B}T_{kin}$. The $\langle\delta
v_{\parallel}^{2}\rangle_{st}\sim 1/M$ scalings in the small and large $M$
limits are observable in Fig. 6($b$) of C. Further, using the definition
$\langle v_{\perp}^{2}\rangle_{st}=\langle{\mathbf{v}}^{2}\rangle_{st}-\langle
v_{\parallel}^{2}\rangle_{st}$ we get

$\displaystyle\langle
v_{\perp}^{2}\rangle_{st}=(d-1)\left[\frac{\lambda^{2}M}{((d-1)M+1)(dM+1)}+\frac{1}{M}\right]$
(20)

and $\langle\delta v_{\perp}^{2}\rangle_{st}=\langle
v_{\perp}^{2}\rangle_{st}$ as $\langle v_{\perp}\rangle=0$. In the overdamped
limit $M\langle v_{\perp}^{2}\rangle=(d-1)$ and in the limit of $M\to\infty$
we find $M\langle v_{\perp}^{2}\rangle=(d-1)[1+\frac{\lambda^{2}}{d(d-1)}]$.
Fig. 6($c$) in C shows the $M^{-1}$ scaling of $\langle v_{\perp}^{2}\rangle$
in the small and large $M$ limits.

###  4.6 Fourth moment of velocity

The calculation of $\langle{\mathbf{v}}^{4}\rangle_{s}$ proceeds as before
using equation(5) and involves the following steps,

$\displaystyle(s+4/M)\langle{\mathbf{v}}^{4}\rangle_{s}={\mathbf{v}}_{0}^{4}+\frac{4\lambda}{M}\langle(\hat{\bm{u}}\cdot{\mathbf{v}}){\mathbf{v}}^{2}\rangle_{s}+\frac{4(d+2)}{M^{2}}\langle{\mathbf{v}}^{2}\rangle_{s},$
(21)
$\displaystyle(s+3/M+(d-1))\langle(\hat{\bm{u}}\cdot{\mathbf{v}}){\mathbf{v}}^{2}\rangle_{s}=(\hat{\bm{u}}_{0}\cdot{\mathbf{v}}_{0}){\mathbf{v}}_{0}^{2}+\frac{\lambda}{M}\langle{\mathbf{v}}^{2}\rangle_{s}+\frac{2\lambda}{M}\langle(\hat{\bm{u}}\cdot{\mathbf{v}})^{2}\rangle_{s}+\frac{2(d+2)}{M^{2}}\langle\hat{\bm{u}}\cdot{\mathbf{v}}\rangle_{s},$

where $\langle\hat{\bm{u}}\cdot{\mathbf{v}}\rangle_{s}$,
$\langle{\mathbf{v}}^{2}\rangle_{s}$ and
$\langle(\hat{\bm{u}}\cdot{\mathbf{v}})^{2}\rangle_{s}$ are already calculated
in equations (8), (11) and (18) respectively. Using them, we get the exact
form of $\langle{\mathbf{v}}^{4}\rangle_{s}$ as shown in A. The inverse
Laplace transform gives the time-dependence
$\langle{\mathbf{v}}^{4}\rangle(t)$ shown in equation(A) of Appendix. Here, we
present the steady-state result,

$\displaystyle\langle{{\mathbf{v}}}^{4}\rangle_{st}=\frac{d(d+2)}{M^{2}}+\frac{2(d+2)\lambda^{2}}{((d-1)M+1)M}+\frac{\lambda^{4}((d+2)M+3)}{((d-1)M+1)((d-1)M+3)(dM+1)}$
(23)

The short-time behavior with initial condition ${\mathbf{v}}_{0}={\bf 0}$ can
be analyzed in terms of a series expansion of
$\langle{\mathbf{v}}^{4}\rangle(t)$ around $t=0$ yielding,

$\displaystyle\langle{\mathbf{v}}^{4}\rangle(t)=\frac{4d(d+2)}{M^{4}}t^{2}+{\cal
A}_{1}t^{3}+{\cal B}_{1}t^{4}+{\cal{O}}(t^{5}),$

where ${\cal A}_{1}=4[(d+2)(\lambda^{2}M-2d)]/M^{5}$, and $3{\cal
B}_{1}M^{6}=[\lambda^{2}M^{2}\\{-4d(d+1)+3\lambda^{2}+8\\}-24(d+2)\lambda^{2}M+28d(d+2)]$.
As shown in Fig. 1($d$), $\langle{\mathbf{v}}^{4}\rangle(t)$ undergoes a
crossover from $\sim t^{2}$ scaling at short times to $\sim t^{3}$ scaling
near $t_{I}=\frac{4d(d+2)}{M^{4}}/{\cal A}_{1}=dM/(\lambda^{2}M-2d)$. Further,
it crosses over to a $\sim t^{4}$ scaling at $t_{II}={\cal A}_{1}/{\cal
B}_{1}$ before saturating to a steady state. Time-dependence of velocity
moments before reaching a steady state depends on the initial condition, a
point illustrated further in C.

Figure 2: The steady-state kurtosis $\cal{K}$ as a function of $M$ at
$\lambda=10$ ($a$) and as a function of activity $\lambda$ at $M=1$ ($b$). The
solid (dashed) line shows the plot in $d=2$ ($d=3$). ($c$) Phase diagram: A
heat map of steady-state kurtosis as a function of activity $\lambda$ and mass
$M$ in $d=2$. Passive and active states are represented by light (yellow) and
dark (black) regions, respectively. ($d$) Heat maps of $P(v_{x},v_{y})$ in the
passive state identified by point ($iv$) in the phase diagram ($c$) and ($e$)
$P(v_{x},v_{y})$ in the active state identified by point ($ii$) in ($c$).

###  4.7 Kurtosis: deviations from the Gaussian process

The fourth moment $\langle{\mathbf{v}}^{4}\rangle$ for a general Gaussian
process with $\langle{\mathbf{v}}\rangle=0$ in terms of lower order moments
can be expressed as [19]

$\displaystyle\mu_{4}:=\left(1+\frac{2}{d}\right)\langle{\mathbf{v}}^{2}\rangle^{2}$
(24)

The departure from Gaussian behavior can be evaluated using the kurtosis

$\displaystyle{\cal K}=\frac{\langle{\mathbf{v}}^{4}\rangle}{\mu_{4}}-1.$ (25)

At steady-state, it has the form

$\displaystyle{\cal
K}=\frac{-2\lambda^{4}M^{2}((4d-1)M+3)}{(d+2)((d-1)M+3)(dM+1)\left(d^{2}M-dM+d+\lambda^{2}M\right)^{2}}.$
(26)

Thus, kurtosis shows a non-monotonic variation with $M$ (Fig.2(a) ). It
vanishes in the two limits of small and large inertia – in the limit of $M\to
0$ as

$\displaystyle{\cal K}\approx-\frac{2\lambda^{4}}{3d^{2}(d+2)}\,M^{2}$ (27)

and in the limit of $M\to\infty$ as ${\cal{K}}\sim M^{-1}$, indicating the two
passive states with asymptotically Gaussian distributions of velocity. For
intermediate $M$ values, ${\cal K}$ becomes negative and reaches a minimum.
This suggests a re-entrance transition from passive to active to passive
behavior with the increase in inertia.

On the other hand, with activity $\lambda$, kurtosis decreases monotonically
to saturate (Fig.2(b) ). For $\lambda\to 0$, it vanishes as
${\cal{K}}\sim\lambda^{4}$ . In the limit of large activity
$\lambda\to\infty$, it saturates to

$\displaystyle{\cal{K}}=-\frac{2(4dM-M+3)}{(d+2)(dM+1)((d-1)M+3)}.$ (28)

The large amplitude of kurtosis characterizes a strongly non-Gaussian velocity
distribution denoting an active state. As shown in Fig.2(d), the velocity
distribution in two dimensions (2d) corresponding to the active state obtained
from numerical simulations displays a distinct ring shape.

###  4.8 Phase diagram

A heat map of equation (26) in Fig.2(c) produces a phase diagram identifying
active (dark, large amplitude of ${\cal K}$) and passive (light, ${\cal
K}\approx 0$) regions in 2d. It clearly shows a re-entrant transition from
passive (Gaussian) to active (non-Gaussian) to passive (Gaussian) with
increasing inertia at a constant activity, e.g., near $\lambda\approx 100$.
The phase diagram in $d=3$ also shows similar behavior (figure not shown).

The re-entrant phase transitions can be understood using the following
heuristic argument. The velocity distribution becomes Gaussian if the
persistence of the heading direction is lost over the inertial relaxation
time, i.e., $\tau\gg\tau_{r}$, which is equivalent to $M\gg 1$. The phase
diagram shows the actual boundary is near $M=1$. At $M<1$, i.e.,
$\tau<\tau_{r}$, persistence dominates; therefore, the velocity distribution
shows maximum on a $(d-1)$-dimensional hypersphere of radius $v=\lambda$, with
the spherical shape emerging due to isotropy in the initial orientation. Also,
the thermal noise thickens the hypersurface shell by $\langle\delta
v^{2}\rangle\approx d/M$. The $v=\lambda$ peak of the distribution gets
unrecognizable when the velocity fluctuation gets larger than the ring size,
$\langle\delta v^{2}\rangle\gg\lambda^{2}$, leading to the criterion
$M<\lambda^{-2}$ for a re-entrance to the passive Gaussian regime. Note that
this criterion is consistent with equation (27).

###  4.9 Probability distributions of velocity in 2d

The Fokker-Planck equation can be directly analyzed in the two limits of small
and large $M$. For $M\ll 1$, i.e., $\tau_{r}\gg\tau$, the heading direction
can be assumed to be fixed at some angle $\theta$. This regime is equivalent
to the dynamics of a particle under external force. Thus, the steady-state
solution is

$\displaystyle
P(v_{x},v_{y},\theta)={\frac{M}{2\pi}}e^{-\frac{M}{2}[(v_{x}-\lambda\cos\theta)^{2}+(v_{y}-\lambda\sin\theta)^{2}]}.$
(29)

The velocity distribution $P(v_{x},v_{y})$ can be obtained by integrating over
all possible $\theta$, $P(v_{x},v_{y})=\int_{0}^{2\pi}d\theta
P(v_{x},v_{y},\theta)$. The marginal distribution,

$\displaystyle
P(v_{x})=\int_{0}^{2\pi}\frac{d\theta}{2\pi}~{}\sqrt{\frac{M}{2\pi}}e^{-\frac{M}{2}(v_{x}-\lambda\cos\theta)^{2}},$
(30)

by symmetry has the same form as
$P(v_{y})=\int_{0}^{2\pi}\frac{d\theta}{2\pi}~{}\sqrt{\frac{M}{2\pi}}e^{-\frac{M}{2}(v_{y}-\lambda\sin\theta)^{2}}$.
As shown in Fig.3($a$), this distribution already describes the small inertia
crossover between the passive Gaussian to active non-Gaussian (bimodal
symmetric) nature, capturing the numerical simulation results at
$M=10^{-4},10^{-2}$ for $\lambda=100$. Thus, the previously proposed
perturbative calculation [23] is not necessary to qualitatively describe the
non-Gaussian feature. Moreover, as the perturbative calculation is valid only
for small $M$ limits, it cannot capture the large $M$ re-entrance to Gaussian
behavior.

At large $M\gg 1$, orientational relaxation leads to effective equilibrium-
like Gaussian distributions with enhanced effective diffusivity $D_{\rm
eff}=1+\lambda^{2}/[d(d-1)]$ (equation (40) in Sec. 5),

$\displaystyle P(v_{x},v_{y})={\frac{M}{2\pi D_{\rm eff}}}e^{-\frac{M}{2D_{\rm
eff}}(v_{x}^{2}+v_{y}^{2})},~{}P(v_{i})=\sqrt{\frac{M}{2\pi D_{\rm
eff}}}e^{\frac{-Mv_{i}^{2}}{2D_{\rm eff}}}.$ (31)

In the last expression, $v_{i}$ denotes both $v_{x}$ and $v_{y}$. This
expression captures the simulation results at $M=10^{2}$ and $\lambda=100$,
thereby describing the crossover to passive Gaussian behavior at large enough
$M$. Thus, equations (30) and (31) together capture the re-entrance observed
as a function of $M$ as evidenced from the numerical simulations shown in
Fig.3.

Figure 3: Velocity distributions in 2d at $\lambda=100$: ($a$) Probability
distributions of components of velocity $v_{i}$ with $i=x,y$. Lines denote
analytic expressions, and points denote simulation results. The small $M$
Gaussian to non-Gaussian crossover is captured by equation (30) showing good
agreement with simulation results at $M=10^{-4}$ and $M=10^{-2}$. The large
$M=10^{2}$ result from simulations agrees with equation (31). Probability
distributions of velocity components parallel $P(v_{\parallel})$ and
perpendicular $P(v_{\perp})$ to the heading direction are shown in ($b$) and
($c$). The distributions denoted by ($i$), ($ii$), and ($iii$) in the legends
correspond to the points marked in Fig. 2($b$) at fixed $\lambda=100$ and
$M=100$, 0.2, $10^{-4}$, respectively. The lines at $M=10^{-4}$ ($iii$) are
plots of equation (32) and that at $M=100$ ($i$) are plots of equation (33).
The lines at intermediate $M=0.2$ ($ii$) are guides to the eye.

Similar arguments as above can be used to describe the probability
distribution of $v_{\parallel}$ and $v_{\perp}$ in the two limits of small and
high inertia. At small $M$ their distributions take the form

$\displaystyle
P(v_{\parallel})=\sqrt{\frac{M}{2\pi}}e^{-\frac{M}{2}(v_{\parallel}-\lambda)^{2}},~{}P(v_{\perp})=\sqrt{\frac{M}{2\pi}}e^{-\frac{M}{2}v_{\perp}^{2}}.$
(32)

These expressions agree with the numerical result at $M=10^{-4}$ (Fig. 3).
Equation (32) also suggests $\langle v_{\parallel}\rangle=\lambda$, $\langle
v_{\perp}\rangle=0$, $\langle\delta v_{\parallel}^{2}\rangle\sim 1/M$ and
$\langle\delta v_{\perp}^{2}\rangle\sim 1/M$ at small $M$, agreeing with
equations (17) and (19). In the large $M$ limit, distributions are described
by the effective diffusion constant,

$\displaystyle P(v_{\parallel})=\sqrt{\frac{M}{2\pi D_{\rm
eff}}}e^{-\frac{M}{2D_{\rm
eff}}v_{\parallel}^{2}},~{}P(v_{\perp})=\sqrt{\frac{M}{2\pi D_{\rm
eff}}}e^{-\frac{M}{2D_{\rm eff}}v_{\perp}^{2}}$ (33)

The above expressions are tested against the numerical result at $M=10^{2}$
and show good agreements (see Fig. 3). Equation (33) suggests $\langle
v_{\parallel}\rangle=0=\langle v_{\perp}\rangle$ and the variance
$\langle\delta v_{\parallel}^{2}\rangle=\langle\delta v_{\perp}^{2}\rangle\sim
1/M$ at the large $M$ limit, agreeing with equations (17) and (19).

The velocity evolution under active drive Eq.(2) can formally be mapped to the
position evolution of an overdamped ABP in a harmonic trap
$dr_{i}=(\lambda_{a}u_{i}-\beta r_{i})dt+\sqrt{2}\,dB_{i}(t)$ [37], with
activity $\lambda_{a}$ and trap stiffness $\beta$. The mapping gives
$M=\beta^{-1}$, $\lambda=\lambda_{a}/\beta$ and the strength of the thermal
noise $\beta^{-2}$. Thus, although the algebra is equivalent, extracting the
physical meaning from such mapping requires care. A further subtlety involves
the coupling of positional evolution to velocity for inertial ABPs, which we
explore next.

##  5 Displacement moments, active diffusivity, and swim pressure

In this section, we calculate moments of displacement vector to explore their
time evolution. We begin by using $\psi={\mathbf{r}}$ and the initial
condition $\langle\psi\rangle_{0}=0$ in equation(5) to get
$\langle{\mathbf{r}}\rangle_{s}=\langle{\mathbf{v}}\rangle_{s}/s$ . Further
using equation(6), we get

$\displaystyle\langle{\mathbf{r}}\rangle_{s}=\frac{{\mathbf{v}}_{0}}{s(s+1/M)}+\frac{(\lambda/M)\hat{\bm{u}}_{0}}{s(s+1/M)(s+d-1)}.$
(34)

Its inverse Laplace transform gives

$\displaystyle\langle{\mathbf{r}}\rangle(t)=M{\mathbf{v}}_{0}+\frac{\lambda\hat{\bm{u}}_{0}}{(d-1)}+\frac{\lambda\hat{\bm{u}}_{0}\,e^{-(d-1)t}}{(d-1)[(d-1)M-1]}-\left({\mathbf{v}}_{0}+\frac{\lambda\hat{\bm{u}}_{0}}{(d-1)M-1}\right)M\,e^{-t/M}$
(35)

In the limit of $M\to 0$ the above expression gives
$\langle{\mathbf{r}}(t)\rangle=\frac{\lambda\hat{\bm{u}}_{0}}{(d-1)}[1-e^{-(d-1)t}]$
in agreement with the evolution of overdamped ABPs [19].

###  5.1 Mean squared displacement (MSD) and asymptotic diffusivity

Let us now consider $\psi={\mathbf{r}}^{2}$ and the initial condition
${\mathbf{r}}(t=0)=0$ in equation(5). It is easy to see
$\langle{\mathbf{v}}\cdot\nabla\psi\rangle_{s}=2\langle{\mathbf{v}}\cdot{\mathbf{r}}\rangle_{s}$,
${\mathbf{v}}\cdot\nabla_{v}\psi=0$, $\hat{\bm{u}}\cdot\nabla_{v}\psi=0$,
$\nabla_{v}^{2}\psi=0$, and $\nabla_{u}^{2}\psi=0$. Thus, equation(5) leads to
$\langle{\mathbf{r}}^{2}\rangle_{s}=2\langle{\mathbf{v}}\cdot{\mathbf{r}}\rangle_{s}/s$.
The calculation of the position-velocity cross-correlation
$\langle{\mathbf{v}}\cdot{\mathbf{r}}\rangle_{s}$ proceeds in the same way
using $\psi={\mathbf{v}}\cdot{\mathbf{r}}$ in equation(5), using initial
condition $\langle\psi\rangle_{0}=0$,
$\langle{\mathbf{v}}\cdot\nabla\psi\rangle_{s}=\langle{\mathbf{v}}^{2}\rangle_{s}$,
$\langle{\mathbf{v}}\cdot\nabla_{v}\psi\rangle_{s}=\langle{\mathbf{v}}\cdot{\mathbf{r}}\rangle_{s}$,
$\langle\hat{\bm{u}}\cdot\nabla_{v}\psi\rangle_{s}=\langle\hat{\bm{u}}\cdot{\mathbf{r}}\rangle_{s}$,
$\nabla_{v}^{2}\psi=0$, and $\nabla_{u}^{2}\psi=0$. This leads to

$\displaystyle(s+1/M)\langle{\mathbf{v}}\cdot{\mathbf{r}}\rangle_{s}=\langle{\mathbf{v}}^{2}\rangle_{s}+\lambda\langle\hat{\bm{u}}\cdot{\mathbf{r}}\rangle_{s}/M.$
(36)

To complete the calculation, one needs to evaluate the position-orientation
cross-correlation $\langle\hat{\bm{u}}\cdot{\mathbf{r}}\rangle_{s}$, using
$\psi=\hat{\bm{u}}\cdot{\mathbf{r}}$ in equation(5) with
$\langle\psi\rangle_{s}=0$,
$\langle{\mathbf{v}}\cdot\nabla\psi\rangle_{s}=\langle\hat{\bm{u}}\cdot{\mathbf{v}}\rangle_{s}$,
${\mathbf{v}}\cdot\nabla_{v}\psi=0$, $\hat{\bm{u}}\cdot\nabla_{v}\psi=0$,
$\nabla_{v}^{2}\psi=0$, and
$\langle\nabla_{u}^{2}\psi\rangle_{s}=-(d-1)\langle\hat{\bm{u}}\cdot{\mathbf{r}}\rangle_{s}$
to obtain

$\displaystyle(s+(d-1))\langle\hat{\bm{u}}\cdot{\mathbf{r}}\rangle_{s}=\langle\hat{\bm{u}}\cdot{\mathbf{v}}\rangle_{s}.$
(37)

Note that $\langle{\mathbf{v}}^{2}\rangle_{s}$ and
$\langle\hat{\bm{u}}\cdot{\mathbf{v}}\rangle_{s}$ were already calculated in
equations (11) and (8). Thus, we find

$\displaystyle\langle{\mathbf{r}}^{2}\rangle_{s}=\frac{2}{s(s+1/M)}\left[\frac{1}{s+2/M}\left({\mathbf{v}}_{0}^{2}+\frac{2d}{sM^{2}}\right.\right.{\left.\left.+\frac{2\lambda\\{(\hat{\bm{u}}_{0}\cdot{\mathbf{v}}_{0})sM+\lambda\\}}{sM\\{1+[s+(d-1)]M\\}}\right)\right.}$
$\displaystyle{\left.+\frac{\lambda\\{(\hat{\bm{u}}_{0}\cdot{\mathbf{v}}_{0})sM+\lambda\\}}{sM(s+(d-1))\\{1+(s+(d-1))M\\}}\right].}$
(38)

The inverse Laplace transform of equation(5.1) gives the full time dependence
of MSD,

$\displaystyle\langle{\mathbf{r}}^{2}\rangle(t)=2t\left(d+\frac{\lambda^{2}}{d-1}\right)+\frac{2\lambda[(\hat{\bm{u}}_{0}\cdot{\mathbf{v}}_{0})(d-1)M-\lambda]}{(d-1)^{2}[(d-1)M-1]}e^{-(d-1)t}$
$\displaystyle-\frac{2\lambda
M\\{(\hat{\bm{u}}_{0}\cdot{\mathbf{v}}_{0})[(d-1)M+1]-\lambda\\}}{(d-1)^{3}M^{2}-d+1}e^{-((d-1)+M^{-1})t}$
$\displaystyle+M\left(-(d-{\mathbf{v}}_{0}^{2}M)+\frac{2(\hat{\bm{u}}_{0}\cdot{\mathbf{v}}_{0})M\lambda-M\lambda^{2}}{(d-1)M-1}\right)e^{-2t/M}$
$\displaystyle+2M\left[(2d-{\mathbf{v}}_{0}^{2}M)+\frac{(\hat{\bm{u}}_{0}\cdot{\mathbf{v}}_{0})(1-2(d-1)M)\lambda}{(d-1)((d-1)M-1)}+\frac{\\{2(d-1)M-1\\}\lambda^{2}}{(d-1)((d-1)M-1)}\right]e^{-t/M}$
$\displaystyle+M({\mathbf{v}}_{0}^{2}M-3d)+\frac{2(\hat{\bm{u}}_{0}\cdot{\mathbf{v}}_{0})\lambda
M}{d-1}-\frac{\lambda^{2}\\{(d-1)M(3(d-1)M+4)+2\\}}{(d-1)^{3}M+(d-1)^{2}}$
(39)

As shown in Fig. 4($a$), the above equation captures the numerical simulation
results of MSD. Moreover, the MSD shows a curious $\sim t^{3}$ scaling at
short times before reaching the asymptotic diffusive scaling. Note that the
long-time diffusive behavior

$\displaystyle\lim_{t\to\infty}\langle{\mathbf{r}}^{2}\rangle(t)$
$\displaystyle=2dD_{\rm eff}t~{}{~{}\rm with~{}~{}}D_{\rm
eff}=1+\lambda^{2}/(d(d-1))$ (40)

is independent of inertia, and the diffusion constant has the same form as
overdamped ABPs [19, 14]. To analyze the short-time behavior, we expand
$\langle{\mathbf{r}}^{2}\rangle(t)$ around $t=0$

$\displaystyle\langle{\mathbf{r}}^{2}\rangle(t)={\mathbf{v}}_{0}^{2}t^{2}+{\cal
A}_{2}t^{3}+{\cal B}_{2}t^{4}+{\cal O}\left(t^{5}\right)$ (41)

where ${\cal
A}_{2}=(-3{\mathbf{v}}_{0}^{2}M+3(\hat{\bm{u}}_{0}\cdot{\mathbf{v}}_{0})\lambda
M+2d)/(3M^{2})$ and ${\cal
B}_{2}=[7{\mathbf{v}}_{0}^{2}M-2(\hat{\bm{u}}_{0}\cdot{\mathbf{v}}_{0})\lambda
M(2(d-1)M+5)-6d+3\lambda^{2}M]/(12M^{3})$. Equation (41) agrees with the
observation of short-time ballistic behavior of MSD [14] if the initial
velocity is drawn from the equilibrium distribution
${\mathbf{v}}_{0}^{2}=d/M$. However, a choice of ${\mathbf{v}}_{0}={\bf 0}$
leads to $\langle{\mathbf{r}}^{2}\rangle\sim t^{3}$ at short times in
agreement with Fig.4, and the series expansion simplifies to

$\displaystyle\langle{\mathbf{r}}^{2}\rangle(t)=\frac{2d}{3M^{2}}t^{3}+\frac{\left(\lambda^{2}M-2d\right)}{4M^{3}}t^{4}-\frac{\left(2\lambda^{2}M^{2}(d-1)+5\lambda^{2}M-7d\right)}{30M^{4}}t^{5}+{\cal
O}\left(t^{6}\right).$ (42)

The expansion explains the short-time crossover from
$\langle{\mathbf{r}}^{2}\rangle\sim t^{3}$ to
$\langle{\mathbf{r}}^{2}\rangle\sim t^{4}$ observed from numerical simulations
near $t_{I}=8dM/[3(\lambda^{2}M-2d)]$ provided $\lambda^{2}M>2d$, a criterion
fulfilled by the lines ($i$) ($M=100,\lambda=100$) and ($ii$)
($M=0.2,\lambda=100$) in Fig. 4($a$). Note that for lines ($iii$)
($M=10^{-4}$, $\lambda=100$) and ($iv$) ($M=0.2$, $\lambda=1$), the criterion
$\lambda^{2}M>2d$ is not obeyed and as a result beyond the initial $\sim
t^{3}$ scaling, $\langle{\mathbf{r}}^{2}\rangle$ directly approaches the
asymptotic $\sim t$ behavior.

Figure 4: Evolution of $\langle\mathbf{r^{2}}\rangle/t$ and
$\langle\mathbf{r}^{4}\rangle/t^{2}$ with time in $d=2$ with initial velocity
${\mathbf{v}}_{0}=(0,0)$. Lines in ($a$) show plots of
$\langle\mathbf{r}^{2}\rangle$ obtained from equation (5.1) and in ($b$) show
plots of $\langle\mathbf{r}^{4}\rangle$ obtained from solving equations
(48)-(59) and taking inverse Laplace transform. The points $(i)$, $(ii)$,
$(iii)$, and $(iv)$ are the same as shown in Fig.2($b$). The points denote
simulation results.

###  5.2 Position orientation cross-correlation and active pressure

While calculating $\langle{\mathbf{r}}^{2}\rangle$, we computed the cross-
correlation $\langle\hat{\bm{u}}\cdot{\mathbf{r}}\rangle$. Here, we consider
its role in determining the active pressure of ABPs. Note that the virial term
for an ideal gas of $N$ ABPs due to activity
$\pi^{a}=(1/d\,V)\sum_{i=1}^{N}\langle{\mathbf{r}}^{\prime}_{i}\cdot\gamma\lambda^{\prime}\hat{\bm{u}}_{i}\rangle$
with $V$ denoting an effective volume [44, 45]. The dimensionless pressure has
the form ${\cal
P}=\rho(\lambda/d)\langle\hat{\bm{u}}\cdot{\mathbf{r}}\rangle_{st}$, where
$\rho$ denotes the dimensionless particle density and
$\langle\hat{\bm{u}}\cdot{\mathbf{r}}\rangle_{st}=\frac{\lambda}{(d-1)(1+(d-1)M)}$.
Thus, the active pressure

$\displaystyle{\cal P}=\rho\frac{\lambda^{2}}{d(d-1)}\frac{1}{[1+(d-1)M]}.$
(43)

Unlike in passive particles or overdamped ABPs, the active swim pressure turns
out to be a function of inertia $M$. Note that ${\cal P}$ can be re-expressed
in terms of the excess diffusivity due to activity $D^{(a)}_{\rm
eff}=\frac{\lambda^{2}}{d(d-1)}$ such that ${\cal P}=\rho D^{(a)}_{\rm
eff}\,\frac{1}{[1+(d-1)M]}$. In the limit of $M\to 0$, swim pressure is the
maximum, ${\cal P}=\rho D^{(a)}_{\rm eff}$. This result in the overdamped
limit agrees with earlier calculations for ABPs in $d=3$ [45]. However, for
inertial ABPs, the swim pressure ${\cal P}$ vanishes as $M^{-1}$ in the limit
of large inertia. The decrease of swim stress with translational inertia was
observed before in Ref. [20, 22].

###  5.3 Fourth moment of displacement

The calculation of the fourth moment in the Laplace space is long but
straightforward. We show the essential steps of this calculation in B. The
inverse Laplace transform can be used to obtain
$\langle{\mathbf{r}}^{4}\rangle(t)$. As the expression is too lengthy to show
here, we plot the expression $\langle{\mathbf{r}}^{4}\rangle(t)$ as a function
of time and compare it with the result of simulation for various $M$ and
$\lambda$ values in Fig.4($b$) to find good agreement. Here, we analyze the
expression in the short and long time limits. The asymptotic behavior of
$\langle{{\mathbf{r}}}^{4}\rangle$ in the long time limit gives

$\displaystyle\lim_{t\to\infty}\langle{{\mathbf{r}}}^{4}\rangle({t})=\frac{4(d+2)\left(d(d-1)+\lambda^{2}\right)^{2}}{d(d-1)^{2}}{t}^{2},$
(44)

with the coefficient of the $t^{2}$ scaling independent of inertia. The
$t^{6}$ scaling in the $t\to 0$ limit observed in Fig.4($b$) can be understood
performing a series expansion of the expression around $t=0$ with initial
velocity ${\mathbf{v}}_{0}={\bf 0}$

$\displaystyle\langle{\mathbf{r}}^{4}\rangle(t)=\frac{4d(d+2)}{9M^{4}}t^{6}+\frac{(d+2)(\lambda^{2}M-2d)}{3M^{5}}t^{7}+{\cal
C}_{3}t^{8}+{\cal{O}}(t^{9}),$ (45)

where ${\cal
C}_{3}=\\{\lambda^{2}M^{2}[45\lambda^{2}-64(d^{2}+d-2)]-340(d+2)\lambda^{2}M+404d(d+2)\\}/(720M^{6})$.
Note that the shortest time $\sim t^{6}$ scaling is due to equilibrium
fluctuations, independent of activity. This behavior with initial velocity
${\mathbf{v}}_{0}={\bf 0}$ starkly contrasts the shortest-time
$\langle{\mathbf{r}}^{4}\rangle(t)\sim t^{2}$ scaling observed for overdamped
ABPs [19]. The short-time $\sim t^{2}$ scaling for overdamped ABPs is
recovered if one keeps a non-zero ${\mathbf{v}}_{0}$ while taking the limit
$M\to 0$. The first non-equilibrium influence comes in
$\langle{\mathbf{r}}^{4}\rangle(t)$ determining the crossover to $\sim t^{7}$
scaling at $t_{I}=4dM/[3(\lambda^{2}M-2d)]$. This crossover will be observable
if $\lambda^{2}M>2d$. As a result, we see this crossover in lines ($i$) and
($ii$) of Fig.4($b$), but not in lines ($iii$) and ($iv$).

##  6 Conclusion

In this paper, we conducted a thorough analysis of the dynamics of free
inertial Active Brownian Particles (ABP) in the presence of thermal noise. To
achieve this, we employed a method based on the Laplace transform of the
Fokker-Planck equation, allowing us to obtain expressions of time-dependent
dynamical moments of arbitrary observables in any $d$-dimensional space. We
obtained explicit expressions for several moments of velocity and displacement
and equal time cross-correlations. The analytical predictions were rigorously
validated through direct numerical simulations in two dimensions.

Our study has uncovered several intriguing characteristics of inertial active
particles, setting them apart from their overdamped and passive counterparts.
Notably, the inertia-dependent characteristics manifested in the steady-state
kinetic temperature and swim pressure while leaving the late-time diffusivity
unchanged. Specifically, the kinetic temperature exhibited an initial increase
before reaching saturation, while the pressure consistently demonstrated a
monotonic decrease.

Furthermore, our study revealed fascinating re-entrant phase transitions in
the steady-state velocity distribution, transitioning from passive Gaussian to
active non-Gaussian behaviors. To illustrate this phenomenon, we constructed a
detailed phase diagram in the activity-inertia plane, utilizing the exact
expression of $d$-dimensional kurtosis. Additionally, we proposed approximate
velocity distribution formulas that are closely aligned with numerical
simulations, especially in the realms of small and large inertia limits. These
formulations also captured the transition from low inertia passive to active
states.

Our exact expressions, beyond describing steady states, could be used to
explain time-dependent crossovers observed in various moments of velocity and
displacement. Overall, this study provides significant insight into the
intricate interplay between activity, inertia, and thermal noise governing the
behavior of inertial ABPs.

## Acknowledgments

DC thanks Abhishek Dhar and Fernando Peruani for collaborations on related
topics and acknowledges research grants from DAE
(1603/2/2020/IoP/R&D-II/150288) and SERB, India (MTR/2019/000750), and thanks
ICTS-TIFR, Bangalore, for an Associateship.

##  Appendix A Fourth moment of velocity

Inserting equations (4.6), (8), (11) and (18) into equation (21), we get an
exact form of the fourth order moment of velocity
$\langle{\mathbf{v}}^{4}\rangle_{s}$ in Laplace space

$\displaystyle\langle{\mathbf{v}}^{4}\rangle_{s}=\frac{8d(d+2)}{M^{4}s(s+2/M)(s+4/M)}+\frac{8(d+2)\lambda^{2}}{M^{4}s(s+2/M)(s+4/M)(s+1/M+d-1)}$
$\displaystyle+\frac{8(d+2)\lambda^{2}}{M^{4}s(s+4/M)(s+3/M+d-1)(s+1/M+d-1)}$
$\displaystyle+\frac{8d\lambda^{2}}{M^{4}s(s+2/M)(s+4/M)(s+3/M+d-1)}+\frac{16\lambda^{2}}{M^{4}s(s+2/M+2d)(s+3/M+d-1)(s+4/M)}$
$\displaystyle+\frac{32d\lambda^{2}}{M^{4}s(s+2/M)(s+4/M)(s+2/M+2d)(s+3/M+d-1)}$
$\displaystyle+\frac{8\lambda^{4}}{M^{4}s(s+2/M)(s+4/M)(s+1/M+d-1)(s+3/M+d-1)}$
$\displaystyle+\frac{16\lambda^{4}}{M^{4}s(s+4/M)(s+1/M+d-1)(s+3/M+d-1)(s+2/M+2d)}$
$\displaystyle+\frac{32\lambda^{4}}{M^{4}s(s+2/M)(s+4/M)(s+1/M+d-1)(s+3/M+d-1)(s+2/M+2d)}$
$\displaystyle+\frac{{\mathbf{v}}_{0}^{4}}{s+4/M}+\frac{8(\hat{\bm{u}}_{0}\cdot{\mathbf{v}}_{0})^{2}\lambda^{2}}{M^{2}(s+4/M)(s+3/M+d-1)(s+2/M+2d)}+\frac{4(d+2){\mathbf{v}}_{0}^{2}}{M^{2}(s+2/M)(s+4/M)}$
$\displaystyle+\frac{4\lambda^{2}{\mathbf{v}}_{0}^{2}}{M^{2}(s+2/M)(s+4/M)(s+3/M+d-1)}$
$\displaystyle+\frac{16\lambda^{2}{\mathbf{v}}_{0}^{2}}{M^{2}(s+2/M)(s+4/M)(s+2/M+2d)(s+3/M+d-1)}$
$\displaystyle+\frac{8(d+2)\lambda(\hat{\bm{u}}_{0}\cdot{\mathbf{v}}_{0})}{M^{3}(s+2/M)(s+4/M)(s+1/M+d-1)}+\frac{8(d+2)\lambda(\hat{\bm{u}}_{0}\cdot{\mathbf{v}}_{0})}{M^{3}(s+4/M)(s+1/M+d-1)(s+3/M+d-1)}$
$\displaystyle+\frac{8\lambda^{3}(\hat{\bm{u}}_{0}\cdot{\mathbf{v}}_{0})}{M^{3}(s+2/M)(s+4/M)(s+1/M+d-1)(s+3/M+d-1)}$
$\displaystyle+\frac{16\lambda^{3}(\hat{\bm{u}}_{0}\cdot{\mathbf{v}}_{0})}{M^{3}(s+4/M)(s+1/M+d-1)(s+3/M+d-1)(s+2/M+2d)}$
$\displaystyle+\frac{32\lambda^{3}(\hat{\bm{u}}_{0}\cdot{\mathbf{v}}_{0})}{M^{3}(s+2/M)(s+4/M)(s+1/M+d-1)(s+3/M+d-1)(s+2/M+2d)}$
$\displaystyle+\frac{4(\hat{\bm{u}}_{0}\cdot{\mathbf{v}}_{0}){\mathbf{v}}_{0}^{2}}{M(s+4/M)(s+3/M+d-1)}$
(46)

The inverse Laplace transform of equation (A) gives an exact form of
$\langle{\mathbf{v}}^{4}\rangle$(t)

$\displaystyle\langle{\mathbf{v}}^{4}(t)\rangle=\frac{d(d+2)}{M^{2}}+\frac{2(d+2)\lambda^{2}}{M+(d-1)M^{2}}+\frac{[(d+2)M+3]\lambda^{4}}{(dM+1)[(d-1)M+1][(d-1)M+3]}$
$\displaystyle+\frac{2(d+2)\\{d^{2}-[d(d-1)+\lambda^{2}]^{2}M^{2}\\}}{M^{2}d[(d-1)^{2}M^{2}-1]}e^{-2t/M}$
$\displaystyle+\frac{e^{-4t/M}}{M^{2}[(d-1)M-3][(d-1)M-1](dM-1)}\left[\\{(d+2)M-3\\}M^{2}\lambda^{4}\right.+$
$\displaystyle
2(d+2)((d-1)M-3)(dM-1)M\lambda^{2}+\left.d(d+2)\\{(d-1)M-3\\}\\{(d-1)M-1\\}(dM-1)\right]$
$\displaystyle+\frac{4(d-1)\lambda^{4}}{d(dM-1)(dM+1)[(d+1)M-1][(d+1)M+1]}e^{-2(d+M^{-1})t}$
$\displaystyle+\frac{4\lambda^{2}\\{(d+2)[(d-1)M-3][(d+1)M+1]+\lambda^{2}[(d-7)M^{2}-3M]\\}}{M[(d-1)M-3][(d-1)M-1][(d-1)M+1][(d+1)M+1]}e^{-((d-1)+M^{-1})t}$
$\displaystyle-\frac{4\lambda^{2}\\{(d+2)[(d-1)M+3][(d+1)M-1]+\lambda^{2}[(d-7)M^{2}+3M]\\}}{M[(d-1)M-1][(d-1)M+1][(d-1)M+3][(d+1)M-1]}e^{-((d-1)+3M^{-1})t}$
$\displaystyle+{\mathbf{v}}_{0}^{4}e^{-4t/M}+\frac{4\lambda(\hat{\bm{u}}_{0}\cdot{\mathbf{v}}_{0}){\mathbf{v}}_{0}^{2}}{((d-1)M-1)}\left[e^{-4t/M}-e^{-((d-1)+3M^{-1})t}\right]$
$\displaystyle+\frac{4(\hat{\bm{u}}_{0}\cdot{\mathbf{v}}_{0})^{2}\lambda^{2}}{(dM-1)(dM+M-1)((d-1)M-1)}\left[((d-1)M-1)e^{-2(d+M^{-1})t}\right.$
$\displaystyle\left.-2(dM-1)e^{-(d-1+3M^{-1})t}+(dM+M-1)e^{-4t/M}\right]$
$\displaystyle+\frac{2(d+2){\mathbf{v}}_{0}^{2}}{M}\left(e^{-2t/M}-e^{-4t/M}\right)+2\lambda^{2}{\mathbf{v}}_{0}^{2}\left(-\frac{2e^{-2(d+M^{-1})t}}{d(dM-1)(dM+M-1)}+\frac{d+2}{d((d-1)M+1)}e^{-2t/M}\right.$
$\displaystyle\left.+\frac{1-(d+2)M}{(dM-1)((d-1)M-1)}e^{-4t/M}+\frac{2(d+5)M-1}{(dM+M-1)((d-1)M+1)((d-1)M-1)}e^{-(d-1+3M^{-1})t}\right)$
$\displaystyle+\lambda(\hat{\bm{u}}_{0}\cdot{\mathbf{v}}_{0})\frac{4(d+2)\left(e^{2t/M}-1\right)e^{-\left(d+4M^{-1}\right)t}\left(e^{dt}-e^{(1+M^{-1})t}\right)}{M((d-1)M-1)}$
$\displaystyle+4\lambda^{3}(\hat{\bm{u}}_{0}\cdot{\mathbf{v}}_{0})\left(\frac{((d-7)M+3)e^{-(d-1+3M^{-1})t}}{((d-1)M-1)((d-1)M+1)(dM+M-1)}\right.$
$\displaystyle\left.-\frac{((d-7)M-3)e^{-((d-1)+M^{-1})t}}{((d-1)M-3)((d-1)M-1)(dM+M+1)}-\frac{2(d-1)e^{-2\left(d+M^{-1}\right)t}}{d(dM-1)(dM+M-1)(dM+M+1)}\right.$
$\displaystyle\left.+\frac{(d+2)e^{-2t/M}}{d\left((d-1)^{2}M^{2}-1\right)}+\frac{(3-(d+2)M)e^{-{4t}/{M}}}{((d-1)M-3)((d-1)M-1)(dM-1)}\right)$
(47)

##  Appendix B Fourth moment of displacement

The calculation of $\langle{\mathbf{r}}^{4}\rangle_{s}$ requires following
steps :

$\displaystyle
s\langle{\mathbf{r}}^{4}\rangle_{s}=4\langle({\mathbf{v}}\cdot{\mathbf{r}}){\mathbf{r}}^{2}\rangle_{s}$
(48)
$\displaystyle(s+1/M)\langle({\mathbf{v}}\cdot{\mathbf{r}}){\mathbf{r}}^{2}\rangle_{s}=\langle{\mathbf{v}}^{2}{\mathbf{r}}^{2}\rangle_{s}+2\langle({\mathbf{v}}\cdot{\mathbf{r}})^{2}\rangle_{s}+(\lambda/M)\langle(\hat{\bm{u}}\cdot{\mathbf{r}}){\mathbf{r}}^{2}\rangle_{s}$
(49)
$\displaystyle(s+2/M)\langle{\mathbf{v}}^{2}{\mathbf{r}}^{2}\rangle_{s}=2\langle({\mathbf{v}}\cdot{\mathbf{r}}){\mathbf{v}}^{2}\rangle_{s}+(2\lambda/M)\langle(\hat{\bm{u}}\cdot{\mathbf{v}}){\mathbf{r}}^{2}\rangle_{s}+(2d/M^{2})\langle{\mathbf{r}}^{2}\rangle_{s}$
(50)
$\displaystyle(s+3/M)\langle({\mathbf{v}}\cdot{\mathbf{r}}){\mathbf{v}}^{2}\rangle_{s}=\langle{\mathbf{v}}^{4}\rangle_{s}+(\lambda/M)\langle(\hat{\bm{u}}\cdot{\mathbf{r}}){\mathbf{v}}^{2}\rangle_{s}+(2\lambda/M)\langle(\hat{\bm{u}}\cdot{\mathbf{v}})({\mathbf{v}}\cdot{\mathbf{r}})\rangle_{s}$
$\displaystyle+2((d+2)/M^{2})\langle{\mathbf{v}}\cdot{\mathbf{r}}\rangle_{s}$
(51)
$\displaystyle(s+2/M+(d-1))\langle(\hat{\bm{u}}\cdot{\mathbf{v}})({\mathbf{v}}\cdot{\mathbf{r}})\rangle_{s}=\langle(\hat{\bm{u}}\cdot{\mathbf{v}}){\mathbf{v}}^{2}\rangle_{s}+(\lambda/M)\langle{\mathbf{v}}\cdot{\mathbf{r}}\rangle_{s}$
$\displaystyle+(\lambda/M)\langle(\hat{\bm{u}}\cdot{\mathbf{v}})(\hat{\bm{u}}\cdot{\mathbf{r}})\rangle_{s}+(2/M^{2})\langle\hat{\bm{u}}\cdot{\mathbf{r}}\rangle_{s}$
(52)
$\displaystyle(s+2/M+(d-1))\langle(\hat{\bm{u}}\cdot{\mathbf{r}}){\mathbf{v}}^{2}\rangle_{s}=\langle(\hat{\bm{u}}\cdot{\mathbf{v}}){\mathbf{v}}^{2}\rangle_{s}+(2\lambda/M)\langle(\hat{\bm{u}}\cdot{\mathbf{v}})(\hat{\bm{u}}\cdot{\mathbf{r}})\rangle_{s}$
$\displaystyle+(2d/M^{2})\langle\hat{\bm{u}}\cdot{\mathbf{r}}\rangle_{s}$ (53)
$\displaystyle(s+1/M+2d)\langle(\hat{\bm{u}}\cdot{\mathbf{v}})(\hat{\bm{u}}\cdot{\mathbf{r}})\rangle_{s}=\langle(\hat{\bm{u}}\cdot{\mathbf{v}})^{2}\rangle_{s}+(\lambda/M)\langle\hat{\bm{u}}\cdot{\mathbf{r}}\rangle_{s}+2\langle{\mathbf{v}}\cdot{\mathbf{r}}\rangle_{s}$
(54)
$\displaystyle(s+1/M+(d-1))\langle(\hat{\bm{u}}\cdot{\mathbf{v}}){\mathbf{r}}^{2}\rangle_{s}=2\langle(\hat{\bm{u}}\cdot{\mathbf{v}})({\mathbf{v}}\cdot{\mathbf{r}})\rangle_{s}+(\lambda/M)\langle{\mathbf{r}}^{2}\rangle_{s}$
(55)
$\displaystyle(s+2/M)\langle({\mathbf{v}}\cdot{\mathbf{r}})^{2}\rangle_{s}=2\langle({\mathbf{v}}\cdot{\mathbf{r}}){\mathbf{v}}^{2}\rangle_{s}+(2\lambda/M)\langle(\hat{\bm{u}}\cdot{\mathbf{r}})({\mathbf{v}}\cdot{\mathbf{r}})\rangle_{s}+(2/M^{2})\langle{\mathbf{r}}^{2}\rangle_{s}$
(56)
$\displaystyle(s+(d-1))\langle(\hat{\bm{u}}\cdot{\mathbf{r}}){\mathbf{r}}^{2}\rangle_{s}=\langle(\hat{\bm{u}}\cdot{\mathbf{v}}){\mathbf{r}}^{2}\rangle_{s}+2\langle(\hat{\bm{u}}\cdot{\mathbf{r}})({\mathbf{v}}\cdot{\mathbf{r}})\rangle_{s}$
(57)
$\displaystyle(s+1/M+(d-1))\langle(\hat{\bm{u}}\cdot{\mathbf{r}})({\mathbf{v}}\cdot{\mathbf{r}})\rangle_{s}=\langle(\hat{\bm{u}}\cdot{\mathbf{v}})({\mathbf{v}}\cdot{\mathbf{r}})\rangle_{s}+\langle(\hat{\bm{u}}\cdot{\mathbf{r}}){\mathbf{v}}^{2}\rangle_{s}$
$\displaystyle+(\lambda/M)\langle(\hat{\bm{u}}\cdot{\mathbf{r}})^{2}\rangle_{s}$
(58)
$\displaystyle(s+2d)\langle(\hat{\bm{u}}\cdot{\mathbf{r}})^{2}\rangle_{s}=2\langle(\hat{\bm{u}}\cdot{\mathbf{v}})(\hat{\bm{u}}\cdot{\mathbf{r}})\rangle_{s}+2\langle{\mathbf{r}}^{2}\rangle_{s}$
(59)

where $\langle{\mathbf{v}}^{2}\rangle_{s}$,
$\langle{\mathbf{v}}^{4}\rangle_{s}$,
$\langle(\hat{\bm{u}}\cdot{\mathbf{v}}){\mathbf{v}}^{2}\rangle_{s}$,
$\langle\hat{\bm{u}}\cdot{\mathbf{v}}\rangle_{s}$,
$\langle(\hat{\bm{u}}\cdot{\mathbf{v}})^{2}\rangle_{s}$,
$\langle{\mathbf{r}}^{2}\rangle_{s}$,
$\langle{\mathbf{v}}\cdot{\mathbf{r}}\rangle_{s}$, and
$\langle\hat{\bm{u}}\cdot{\mathbf{r}}\rangle_{s}$ were already calculated in
equations (11), (21), (4.6), (8), (18), (5.1), (36), and (37) respectively.
Inserting these back in equation(48) one gets
$\langle{\mathbf{r}}^{4}\rangle_{s}$.

##  Appendix C Velocity moments

Figure 5: Evolution of moment of velocity in $d=2$ dimension with
$\lambda=100$, initial velocity ${\mathbf{v}}_{0}=(6,0)$ and initial
orientation $\hat{\bm{u}}_{0}=(1,0)$. Results for two $M$ values from
simulations (points) and analytic calculations (lines) are shown.

In this section, we supplement the results presented in the main text
containing various velocity moments. In Fig. 5, we show a comparison between
analytic and numerical simulation results with initial conditions
${\mathbf{v}}_{0}=(6,0)$, $\hat{\bm{u}}_{0}=(1,0)$. An expansion around $t=0$
using $\hat{\bm{u}}_{0}\cdot{\mathbf{v}}_{0}=v_{0}$ and
${\mathbf{v}}_{0}^{2}=v_{0}^{2}$ gives

$\displaystyle\langle{\mathbf{v}}^{2}\rangle(t)=v_{0}^{2}+\frac{2[d+Mv_{0}(\lambda-
v_{0})]}{M^{2}}t+{\cal{C}}_{4}t^{2}+{\cal{O}}(t^{3}),$ (60)

and

$\displaystyle\langle{\mathbf{v}}^{4}\rangle(t)=v_{0}^{4}+\frac{4v_{0}^{2}[2+d+Mv_{0}(\lambda-
v_{0})]}{M^{2}}t+{\cal{C}}_{5}t^{2}+{\cal{O}}(t^{3}),$ (61)

with $M^{3}{\cal{C}}_{4}=\\{\lambda^{2}M-2d+M\lambda
v_{0}[-3-(d-1)M]+2Mv_{0}^{2}$},
$M^{4}{\cal{C}}_{5}=\\{4d(d+2)-12(d+2)Mv_{0}^{2}+8M^{2}v_{0}^{4}+2M\lambda
v_{0}[4(d+2)+Mv_{0}^{2}(-7-(d-1)M)]+6M^{2}\lambda^{2}v_{0}^{2}\\}$.

Figure 6: Inertial dependence of velocity components: $\langle
v_{\parallel}\rangle_{st}$, $\langle\delta v_{\parallel}^{2}\rangle_{st}$ and
$\langle\delta v_{\perp}^{2}\rangle_{st}$ with $M$ at $\lambda=100$ and $d=2$.
The solid lines are the plots of equations (17), (19), and (20) and the points
denote simulation results.

These relations help to understand the differences in short-time scaling
behaviors from the ${\mathbf{v}}_{0}={\bf 0}$ case illustrated in the main
text. Here, in the limit $t\to 0$, $\langle{\mathbf{v}}^{2}\rangle=v_{0}^{2}$
and $\langle{\mathbf{v}}^{4}\rangle=v_{0}^{4}$, as one would expect. The first
$t$-dependence in $\langle{\mathbf{v}}^{2}\rangle$ and
$\langle{\mathbf{v}}^{4}\rangle$ both show $\sim t$ scaling. Note the
difference from ${\mathbf{v}}_{0}={\bf 0}$ case particularly for
$\langle{\mathbf{v}}^{4}\rangle$, as the linear in $t$ term vanishes for
$v_{0}=0$ in equation (61). Further, as shown in Fig. 5, $\langle
v_{\parallel}\rangle$ can even decrease with time before reaching the steady
state.

Fig. 6 illustrates the steady-state variation of $\langle
v_{\parallel}\rangle_{st}$, $\langle\delta v_{\parallel}^{2}\rangle_{st}$ and
$\langle\delta v_{\perp}^{2}\rangle_{st}$ with $M$ in 2d using $\lambda=100$.
The lines are analytic functions, and the points denote simulation results. As
Fig. 6($a$) shows, $\langle v_{\parallel}\rangle_{st}$ decreases from
$\lambda$ to $0$ monotonically, with increasing mass. Also, the $\langle\delta
v_{\parallel}^{2}\rangle_{st}\sim 1/M$ and $\langle\delta
v_{\perp}^{2}\rangle_{st}\sim 1/M$ scalings in the small and large $M$ limits
are observable in Fig. 6.

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