PandaX Collaboration

# A First Search for Solar ${}^{8}B$ Neutrino in the PandaX-4T Experiment
using Neutrino-Nucleus Coherent Scattering

Wenbo Ma  School of Physics and Astronomy, Shanghai Jiao Tong University, Key
Laboratory for Particle Astrophysics and Cosmology (MoE), Shanghai Key
Laboratory for Particle Physics and Cosmology, Shanghai 200240, China
Abdusalam Abdukerim  School of Physics and Astronomy, Shanghai Jiao Tong
University, Key Laboratory for Particle Astrophysics and Cosmology (MoE),
Shanghai Key Laboratory for Particle Physics and Cosmology, Shanghai 200240,
China  Chen Cheng  School of Physics, Sun Yat-Sen University, Guangzhou
510275, China  Zihao Bo  School of Physics and Astronomy, Shanghai Jiao Tong
University, Key Laboratory for Particle Astrophysics and Cosmology (MoE),
Shanghai Key Laboratory for Particle Physics and Cosmology, Shanghai 200240,
China  Wei Chen  School of Physics and Astronomy, Shanghai Jiao Tong
University, Key Laboratory for Particle Astrophysics and Cosmology (MoE),
Shanghai Key Laboratory for Particle Physics and Cosmology, Shanghai 200240,
China  Xun Chen  School of Physics and Astronomy, Shanghai Jiao Tong
University, Key Laboratory for Particle Astrophysics and Cosmology (MoE),
Shanghai Key Laboratory for Particle Physics and Cosmology, Shanghai 200240,
China  Shanghai Jiao Tong University Sichuan Research Institute, Chengdu
610213, China  Yunhua Chen  Yalong River Hydropower Development Company, Ltd.,
288 Shuanglin Road, Chengdu 610051, China  Zhaokan Cheng  Sino-French
Institute of Nuclear Engineering and Technology, Sun Yat-Sen University,
Zhuhai, 519082, China  Xiangyi Cui  Tsung-Dao Lee Institute, Shanghai Jiao
Tong University, Shanghai, 200240, China  Yingjie Fan  School of Physics,
Nankai University, Tianjin 300071, China  Deqing Fang  Key Laboratory of
Nuclear Physics and Ion-beam Application (MOE), Institute of Modern Physics,
Fudan University, Shanghai 200433, China  Changbo Fu  Key Laboratory of
Nuclear Physics and Ion-beam Application (MOE), Institute of Modern Physics,
Fudan University, Shanghai 200433, China  Mengting Fu  School of Physics,
Peking University, Beijing 100871, China  Lisheng Geng  School of Physics,
Beihang University, Beijing 102206, China  International Research Center for
Nuclei and Particles in the Cosmos & Beijing Key Laboratory of Advanced
Nuclear Materials and Physics, Beihang University, Beijing 100191, China
School of Physics and Microelectronics, Zhengzhou University, Zhengzhou, Henan
450001, China  Karl Giboni  School of Physics and Astronomy, Shanghai Jiao
Tong University, Key Laboratory for Particle Astrophysics and Cosmology (MoE),
Shanghai Key Laboratory for Particle Physics and Cosmology, Shanghai 200240,
China  Linhui Gu  School of Physics and Astronomy, Shanghai Jiao Tong
University, Key Laboratory for Particle Astrophysics and Cosmology (MoE),
Shanghai Key Laboratory for Particle Physics and Cosmology, Shanghai 200240,
China  Xuyuan Guo  Yalong River Hydropower Development Company, Ltd., 288
Shuanglin Road, Chengdu 610051, China  Chencheng Han  Tsung-Dao Lee Institute,
Shanghai Jiao Tong University, Shanghai, 200240, China  Ke Han  School of
Physics and Astronomy, Shanghai Jiao Tong University, Key Laboratory for
Particle Astrophysics and Cosmology (MoE), Shanghai Key Laboratory for
Particle Physics and Cosmology, Shanghai 200240, China  Changda He  School of
Physics and Astronomy, Shanghai Jiao Tong University, Key Laboratory for
Particle Astrophysics and Cosmology (MoE), Shanghai Key Laboratory for
Particle Physics and Cosmology, Shanghai 200240, China  Jinrong He  Yalong
River Hydropower Development Company, Ltd., 288 Shuanglin Road, Chengdu
610051, China  Di Huang  School of Physics and Astronomy, Shanghai Jiao Tong
University, Key Laboratory for Particle Astrophysics and Cosmology (MoE),
Shanghai Key Laboratory for Particle Physics and Cosmology, Shanghai 200240,
China  Yanlin Huang  School of Medical Instrument and Food Engineering,
University of Shanghai for Science and Technology, Shanghai 200093, China
Zhou Huang  School of Physics and Astronomy, Shanghai Jiao Tong University,
Key Laboratory for Particle Astrophysics and Cosmology (MoE), Shanghai Key
Laboratory for Particle Physics and Cosmology, Shanghai 200240, China  Ruquan
Hou  Shanghai Jiao Tong University Sichuan Research Institute, Chengdu 610213,
China  Xiangdong Ji  Department of Physics, University of Maryland, College
Park, Maryland 20742, USA  Yonglin Ju  School of Mechanical Engineering,
Shanghai Jiao Tong University, Shanghai 200240, China  Chenxiang Li  School of
Physics and Astronomy, Shanghai Jiao Tong University, Key Laboratory for
Particle Astrophysics and Cosmology (MoE), Shanghai Key Laboratory for
Particle Physics and Cosmology, Shanghai 200240, China  Jiafu Li  School of
Physics, Sun Yat-Sen University, Guangzhou 510275, China  Mingchuan Li  Yalong
River Hydropower Development Company, Ltd., 288 Shuanglin Road, Chengdu
610051, China  Shu Li  School of Mechanical Engineering, Shanghai Jiao Tong
University, Shanghai 200240, China  Shuaijie Li  Tsung-Dao Lee Institute,
Shanghai Jiao Tong University, Shanghai, 200240, China  Qing Lin
<EMAIL_ADDRESS>State Key Laboratory of Particle Detection and
Electronics, University of Science and Technology of China, Hefei 230026,
China  Department of Modern Physics, University of Science and Technology of
China, Hefei 230026, China  Jianglai Liu<EMAIL_ADDRESS>School of
Physics and Astronomy, Shanghai Jiao Tong University, Key Laboratory for
Particle Astrophysics and Cosmology (MoE), Shanghai Key Laboratory for
Particle Physics and Cosmology, Shanghai 200240, China  Tsung-Dao Lee
Institute, Shanghai Jiao Tong University, Shanghai, 200240, China  Shanghai
Jiao Tong University Sichuan Research Institute, Chengdu 610213, China
Xiaoying Lu  Research Center for Particle Science and Technology, Institute of
Frontier and Interdisciplinary Science, Shandong University, Qingdao 266237,
Shandong, China  Key Laboratory of Particle Physics and Particle Irradiation
of Ministry of Education, Shandong University, Qingdao 266237, Shandong, China
Lingyin Luo  School of Physics, Peking University, Beijing 100871, China
Yunyang Luo  Department of Modern Physics, University of Science and
Technology of China, Hefei 230026, China  Yugang Ma  Key Laboratory of Nuclear
Physics and Ion-beam Application (MOE), Institute of Modern Physics, Fudan
University, Shanghai 200433, China  Yajun Mao  School of Physics, Peking
University, Beijing 100871, China  Nasir Shaheed  Research Center for Particle
Science and Technology, Institute of Frontier and Interdisciplinary Science,
Shandong University, Qingdao 266237, Shandong, China  Key Laboratory of
Particle Physics and Particle Irradiation of Ministry of Education, Shandong
University, Qingdao 266237, Shandong, China  Yue Meng<EMAIL_ADDRESS>School of Physics and Astronomy, Shanghai Jiao Tong University, Key Laboratory
for Particle Astrophysics and Cosmology (MoE), Shanghai Key Laboratory for
Particle Physics and Cosmology, Shanghai 200240, China  Shanghai Jiao Tong
University Sichuan Research Institute, Chengdu 610213, China  Xuyang Ning
School of Physics and Astronomy, Shanghai Jiao Tong University, Key Laboratory
for Particle Astrophysics and Cosmology (MoE), Shanghai Key Laboratory for
Particle Physics and Cosmology, Shanghai 200240, China  Ningchun Qi  Yalong
River Hydropower Development Company, Ltd., 288 Shuanglin Road, Chengdu
610051, China  Zhicheng Qian  School of Physics and Astronomy, Shanghai Jiao
Tong University, Key Laboratory for Particle Astrophysics and Cosmology (MoE),
Shanghai Key Laboratory for Particle Physics and Cosmology, Shanghai 200240,
China  Xiangxiang Ren  Research Center for Particle Science and Technology,
Institute of Frontier and Interdisciplinary Science, Shandong University,
Qingdao 266237, Shandong, China  Key Laboratory of Particle Physics and
Particle Irradiation of Ministry of Education, Shandong University, Qingdao
266237, Shandong, China  Changsong Shang  Yalong River Hydropower Development
Company, Ltd., 288 Shuanglin Road, Chengdu 610051, China  Xiaofeng Shang
School of Physics and Astronomy, Shanghai Jiao Tong University, Key Laboratory
for Particle Astrophysics and Cosmology (MoE), Shanghai Key Laboratory for
Particle Physics and Cosmology, Shanghai 200240, China  Guofang Shen  School
of Physics, Beihang University, Beijing 102206, China  Lin Si  School of
Physics and Astronomy, Shanghai Jiao Tong University, Key Laboratory for
Particle Astrophysics and Cosmology (MoE), Shanghai Key Laboratory for
Particle Physics and Cosmology, Shanghai 200240, China  Wenliang Sun  Yalong
River Hydropower Development Company, Ltd., 288 Shuanglin Road, Chengdu
610051, China  Andi Tan  Department of Physics, University of Maryland,
College Park, Maryland 20742, USA  Yi Tao  School of Physics and Astronomy,
Shanghai Jiao Tong University, Key Laboratory for Particle Astrophysics and
Cosmology (MoE), Shanghai Key Laboratory for Particle Physics and Cosmology,
Shanghai 200240, China  Shanghai Jiao Tong University Sichuan Research
Institute, Chengdu 610213, China  Anqing Wang  Research Center for Particle
Science and Technology, Institute of Frontier and Interdisciplinary Science,
Shandong University, Qingdao 266237, Shandong, China  Key Laboratory of
Particle Physics and Particle Irradiation of Ministry of Education, Shandong
University, Qingdao 266237, Shandong, China  Meng Wang  Research Center for
Particle Science and Technology, Institute of Frontier and Interdisciplinary
Science, Shandong University, Qingdao 266237, Shandong, China  Key Laboratory
of Particle Physics and Particle Irradiation of Ministry of Education,
Shandong University, Qingdao 266237, Shandong, China  Qiuhong Wang  Key
Laboratory of Nuclear Physics and Ion-beam Application (MOE), Institute of
Modern Physics, Fudan University, Shanghai 200433, China  Shaobo Wang  School
of Physics and Astronomy, Shanghai Jiao Tong University, Key Laboratory for
Particle Astrophysics and Cosmology (MoE), Shanghai Key Laboratory for
Particle Physics and Cosmology, Shanghai 200240, China  SJTU Paris Elite
Institute of Technology, Shanghai Jiao Tong University, Shanghai, 200240,
China  Siguang Wang  School of Physics, Peking University, Beijing 100871,
China  Wei Wang  Sino-French Institute of Nuclear Engineering and Technology,
Sun Yat-Sen University, Zhuhai, 519082, China  School of Physics, Sun Yat-Sen
University, Guangzhou 510275, China  Xiuli Wang  School of Mechanical
Engineering, Shanghai Jiao Tong University, Shanghai 200240, China  Zhou Wang
School of Physics and Astronomy, Shanghai Jiao Tong University, Key Laboratory
for Particle Astrophysics and Cosmology (MoE), Shanghai Key Laboratory for
Particle Physics and Cosmology, Shanghai 200240, China  Shanghai Jiao Tong
University Sichuan Research Institute, Chengdu 610213, China  Tsung-Dao Lee
Institute, Shanghai Jiao Tong University, Shanghai, 200240, China  Yuehuan Wei
Sino-French Institute of Nuclear Engineering and Technology, Sun Yat-Sen
University, Zhuhai, 519082, China  Mengmeng Wu  School of Physics, Sun Yat-Sen
University, Guangzhou 510275, China  Weihao Wu  School of Physics and
Astronomy, Shanghai Jiao Tong University, Key Laboratory for Particle
Astrophysics and Cosmology (MoE), Shanghai Key Laboratory for Particle Physics
and Cosmology, Shanghai 200240, China  Jingkai Xia  School of Physics and
Astronomy, Shanghai Jiao Tong University, Key Laboratory for Particle
Astrophysics and Cosmology (MoE), Shanghai Key Laboratory for Particle Physics
and Cosmology, Shanghai 200240, China  Mengjiao Xiao  Department of Physics,
University of Maryland, College Park, Maryland 20742, USA  Xiang Xiao  School
of Physics, Sun Yat-Sen University, Guangzhou 510275, China  Pengwei Xie
Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai, 200240,
China  Binbin Yan  School of Physics and Astronomy, Shanghai Jiao Tong
University, Key Laboratory for Particle Astrophysics and Cosmology (MoE),
Shanghai Key Laboratory for Particle Physics and Cosmology, Shanghai 200240,
China  Xiyu Yan  School of Medical Instrument and Food Engineering, University
of Shanghai for Science and Technology, Shanghai 200093, China  Jijun Yang
School of Physics and Astronomy, Shanghai Jiao Tong University, Key Laboratory
for Particle Astrophysics and Cosmology (MoE), Shanghai Key Laboratory for
Particle Physics and Cosmology, Shanghai 200240, China  Yong Yang  School of
Physics and Astronomy, Shanghai Jiao Tong University, Key Laboratory for
Particle Astrophysics and Cosmology (MoE), Shanghai Key Laboratory for
Particle Physics and Cosmology, Shanghai 200240, China  Chunxu Yu  School of
Physics, Nankai University, Tianjin 300071, China  Jumin Yuan  Research Center
for Particle Science and Technology, Institute of Frontier and
Interdisciplinary Science, Shandong University, Qingdao 266237, Shandong,
China  Key Laboratory of Particle Physics and Particle Irradiation of Ministry
of Education, Shandong University, Qingdao 266237, Shandong, China  Ying Yuan
School of Physics and Astronomy, Shanghai Jiao Tong University, Key Laboratory
for Particle Astrophysics and Cosmology (MoE), Shanghai Key Laboratory for
Particle Physics and Cosmology, Shanghai 200240, China  Zhe Yuan  Key
Laboratory of Nuclear Physics and Ion-beam Application (MOE), Institute of
Modern Physics, Fudan University, Shanghai 200433, China  Xinning Zeng  School
of Physics and Astronomy, Shanghai Jiao Tong University, Key Laboratory for
Particle Astrophysics and Cosmology (MoE), Shanghai Key Laboratory for
Particle Physics and Cosmology, Shanghai 200240, China  Dan Zhang  Department
of Physics, University of Maryland, College Park, Maryland 20742, USA  Minzhen
Zhang  School of Physics and Astronomy, Shanghai Jiao Tong University, Key
Laboratory for Particle Astrophysics and Cosmology (MoE), Shanghai Key
Laboratory for Particle Physics and Cosmology, Shanghai 200240, China  Peng
Zhang  Yalong River Hydropower Development Company, Ltd., 288 Shuanglin Road,
Chengdu 610051, China  Shibo Zhang  School of Physics and Astronomy, Shanghai
Jiao Tong University, Key Laboratory for Particle Astrophysics and Cosmology
(MoE), Shanghai Key Laboratory for Particle Physics and Cosmology, Shanghai
200240, China  Shu Zhang  School of Physics, Sun Yat-Sen University, Guangzhou
510275, China  Tao Zhang  School of Physics and Astronomy, Shanghai Jiao Tong
University, Key Laboratory for Particle Astrophysics and Cosmology (MoE),
Shanghai Key Laboratory for Particle Physics and Cosmology, Shanghai 200240,
China  Yingxin Zhang  Research Center for Particle Science and Technology,
Institute of Frontier and Interdisciplinary Science, Shandong University,
Qingdao 266237, Shandong, China  Key Laboratory of Particle Physics and
Particle Irradiation of Ministry of Education, Shandong University, Qingdao
266237, Shandong, China  Yuanyuan Zhang  Tsung-Dao Lee Institute, Shanghai
Jiao Tong University, Shanghai, 200240, China  Li Zhao  School of Physics and
Astronomy, Shanghai Jiao Tong University, Key Laboratory for Particle
Astrophysics and Cosmology (MoE), Shanghai Key Laboratory for Particle Physics
and Cosmology, Shanghai 200240, China  Qibin Zheng  School of Medical
Instrument and Food Engineering, University of Shanghai for Science and
Technology, Shanghai 200093, China  Jifang Zhou  Yalong River Hydropower
Development Company, Ltd., 288 Shuanglin Road, Chengdu 610051, China  Ning
Zhou  School of Physics and Astronomy, Shanghai Jiao Tong University, Key
Laboratory for Particle Astrophysics and Cosmology (MoE), Shanghai Key
Laboratory for Particle Physics and Cosmology, Shanghai 200240, China
Xiaopeng Zhou  School of Physics, Beihang University, Beijing 102206, China
Yong Zhou  Yalong River Hydropower Development Company, Ltd., 288 Shuanglin
Road, Chengdu 610051, China  Yubo Zhou  School of Physics and Astronomy,
Shanghai Jiao Tong University, Key Laboratory for Particle Astrophysics and
Cosmology (MoE), Shanghai Key Laboratory for Particle Physics and Cosmology,
Shanghai 200240, China

###### Abstract

A search for interactions from solar 8B neutrinos elastically scattering off
xenon nuclei using PandaX-4T commissioning data is reported. The energy
threshold of this search is further lowered compared with the previous search
for dark matter, with various techniques utilized to suppress the background
that emerges from data with the lowered threshold. A blind analysis is
performed on the data with an effective exposure of 0.48 tonne$\cdot$year, and
no significant excess of events is observed. Among results obtained using the
neutrino-nucleus coherent scattering, our results give the best constraint on
the solar 8B neutrino flux. We further provide a more stringent limit on the
cross section between dark matter and nucleon in the mass range from 3 to 9
GeV/c2.

Due to complex fusion processes inside the Sun, neutrinos are continuously
generated in large amount. As liquid xenon (LXe) detectors dedicated to dark
matter (DM) direct search [aprile2020projected, akerib2020projected,
zhang2019dark] have been developed into the multi-tonne scale in recent years,
they are now able to reach the sensitivity to detect solar neutrinos via
coherent elastic nuclear scattering (CE$\nu$NS). Among all sources of solar
neutrinos, neutrinos produced in the $\beta$ decay of 8B are the most likely
ones to be detected due to the 15 MeV Q value. The flux of 8B solar neutrinos
on Earth has been measured to be approximately 5$\times$106 cm-2s-1
[agostini2018comprehensive, aharmim2013combined], and its CE$\nu$NS has an
energy spectrum hardly distinguishable from that of a 6 GeV/c2 DM particle in
LXe. No experimental determination of the solar neutrino flux using its
CE$\nu$NS signal has been made yet. Recently, the XENON1T collaboration has
published a search for the 8B CE$\nu$NS signal using 0.6 tonne$\cdot$year data
with no excess found [aprile2021search]. Due to the low nuclear recoil (NR)
energy from the 8B CE$\nu$NS, it is crucial to lower the energy threshold. In
this letter, we report a search for CE$\nu$NS induced by the solar 8B
neutrinos using the commissioning data of PandaX-4T (Run0) based on a blind
analysis, with a dedicated data selection, which lowered the energy threshold
(defined as the energy having signal acceptance of 1%) from 1.33 to 0.95 keV.

PandaX-4T dark matter direct search experiment is located in China Jinping
underground Laboratory (CJPL) [kang2010status, li2015second]. the PandaX-4T
experiment utilizes a dual-phase xenon time projection chamber (TPC) with a
sensitive volume of 3.7 tonne of LXe, and two arrays of photo-multipliers
(PMTs) on the top and bottom of the TPC, consisting of 169 and 199 Hamamatsu
3-inch R11410-23 PMTs, respectively. Both the primary scintillation ($S1$) and
the delayed proportional scintillation from drifted electrons ($S2$) of an
event are collected by the PMTs, allowing 3-D position reconstruction with a
resolution of about a few millimeter for $S2$s of $\sim$100 photoelectron (PE)
on the longitudinal and transverse directions, based on the time difference
between the $S1$ and $S2$, and the PMT pattern of the $S2$, respectively. The
waveforms of the PMTs are digitized by CAEN V1725 digitizers and read out
under the self-trigger mode when the pulse amplitude is approximately 1/3 PE
above the baseline [he2021500]. More details of the detector apparatus can be
found in Refs. [meng2021dark, zhao2021cryogenics, he2021500]. PandaX-4T has
reported the most stringent constraint on the spin-independent cross sections
between the nucleon and DM with the DM mass from 5 GeV/c2 to 10 TeV/c2
[meng2021dark] using the 0.63-tonne-year data from Run0.

Compared with the search reported in Ref. [meng2021dark], new data selections
are developed to enhance the detection efficiency and to minimize the extra
background that emerged from data. Thresholds of the $S1$ and $S2$ are lowered
to 0.3 PE and 65 PE (both in charge), respectively, as compared with the 2 PE
and 80 PE in Ref. [meng2021dark]. The systematics of the background and the
energy reconstruction at such low threshold form the core of this analysis.
With these thresholds, two sets of data used in Ref. [meng2021dark] with a
total live time of about 7.5 days show a higher noise rate, likely due to
micro-discharging in the TPC, and are removed from this analysis. The data
selection cuts used in this analysis are described as follows. We adopt four
selection cuts from the previous analysis [meng2021dark], the diffusion cut
($S2$ widths compatible with the expected fluctuation on the electron arrival
time), the veto PMT cut (no signal in the PMTs outside the field cage), the
fiducial volume cut (FV, 2.67 tonnes), and the single scatter cut (only one
$S2$ above 50 PE in the 1-ms event window). Events with large signals are
observed to be followed by small afterglow signals in PandaX-4T and other
experiments [aprile2014observation, akerib2020investigation]. These afterglow
signals usually are single electrons (SEs) which have a strong correlation
with the previous large $S2$ in both time and position. Compared with Ref.
[meng2021dark], a more stringent afterglow veto based on the time and position
difference to the previous event is implemented. Events with a time difference
to previous $S2$ ($>$2000 PE) less than 50 ms or position difference smaller
than 100 mm are excluded. In addition, we veto the event unless the total
charge per unit time and the number of $S1$s in the preceding 1-ms window have
returned back to normal. The afterglow veto cut also includes a set of
“activity” requirements on an event waveform, that the ratio between the main
$S2$ charge and the total event charge
$\mathcal{F}_{S2}$=$q_{S2}/q_{\textrm{event}}>5/6-150/q_{\textrm{event}}$, the
integrated charge in the preceding event window to be less than 20 PE, and the
main $S1$ to be the only signal within 4-$\mu$s around it. The effective live
time of this analysis is estimated to be 64.7 days.

The signal expectation in this analysis is produced by a two-step simulation.
The first step is the same as in Ref. [meng2021dark], in which the correlated
distribution in $S1$ and $S2$ are produced according to a fit to the
calibration data, later referred to as the signal model. In the second step, a
dedicated waveform simulation (WS) is developed. The waveform of the $S1$ is
assembled using sampled $S1$ hits from the neutron calibration data, similar
to the procedure in Ref. [akerib2021simulations]. The waveform of the $S2$ at
any given position is assembled using individual SE waveforms from the data,
with the reconstructed position within a 40-mm radius circle. The width of the
overall assembled waveform at a given depth in the TPC is required to satisfy
the diffusion relation observed from the data. Effects of PMT afterpulsing,
delayed electrons [sorensen2017two, sorensen2017electron,
akerib2020investigation, akerib2021improving], and photo-ionization of
impurities after a large $S2$ are implemented in the WS according to the data.
More details can be found in the appendix.

Figure 1:  Total efficiency (red solid line) to the solar 8B neutrino
CE$\nu$NS in this analysis with the number of $S1$ hits to be 2 or 3. The
blue, green, cyan, and magenta solid lines represent the signal efficiencies
due to the signal reconstruction, data selection, ROI, and BDT, respectively.
The signal efficiency in the previous study [meng2021dark] is also given in
the pink dashed line as a reference. The ideal spectra of the solar 8B
CE$\nu$NS and the DM-nucleus interaction with the DM mass of 4 (8) GeV/c2 with
an assumed DM-nucleon cross section of 10-44 cm2 are overlaid as well in the
black solid and grey dashed (dotted) curves, respectively, with the scale
indicated on the right axis.

The total efficiency to the 8B CE$\nu$NS consists of four components (see Fig.
1): 1) the signal reconstruction, 2) the data selections discussed two
paragraphs earlier, 3) the region-of-interest (ROI), and 4) a cut based on
boosted decision tree (BDT, see later text). the signal reconstruction
includes clustering of PMT hits into signal pulses, classification of the
signal pulses into $S1$s and $S2$s, and pairing of the classified $S1$s and
$S2$s into incident events. Each step of the signal reconstruction is affected
by the presence of dark noises and stray electrons. For the ROI, we require
the number of coincident PMT hits in an $S1$ to be either 2 or 3 in this
analysis. the events with only a single-hit $S1$ are mostly accidental
background originating from the PMT dark noises, and are excluded from the ROI
due to a poor signal-to-background ratio. The $S2$ charge range, uncorrected
for spatial dependence, is further optimized to be 65$-$230 PE for 2-hit $S1$
and 65$-$190 PE for 3-hit $S1$ based on the expected signal-to-background
ratio. This ROI requirement has dominating effects on the signal efficiency.
The efficiencies of 1), 2) and 3) are estimated using the WS and validated by
the neutron calibration data, with their fractional difference (14%) taken as
the systematic uncertainty.

We take the calculated deposit energy spectrum of the solar 8B CE$\nu$NS in
LXe from Ref. [ruppin2014complementarity], which is shown in Fig. 1. The
signal model implements the light and charge production in LXe following the
NEST v2.3.6 parametrization [szydagis2018noble], and the response of signal
detection in the PandaX-4T detector, similar to Ref. [meng2021dark]. The light
and charge yields are extrapolated from the one used in Ref. [meng2021dark],
which has its model parameters fit to the neutron calibration data in the
energy region of the DM search (see Fig. 2). We adopt the relative
uncertainties of the light and charge yields from NEST [szydagis2022noble],
which is based on a global fit to all available measurements, and
conservatively assume them to be uncorrelated.

Figure 2:  Comparison between the light (top panel) and charge (bottom panel)
yields used in this analysis (in the solid black lines) with the nominal NEST
v2.3.6 [szydagis2018noble] (dashed black lines) and other measurements taken
at different drift electric fields [akerib2016low, huang2020ultra,
aprile2018simultaneous, lenardo2019measurement]. The 1$\sigma$ uncertainty
from Ref. [szydagis2022noble] for the light and charge yields are shown in the
grey bands.

The background composition is the same as Ref. [meng2021dark]. With loosened
$S1$ and $S2$ selections, the accidental coincidence (AC) background increases
significantly in comparison to Ref. [meng2021dark], which dominates the
overall background. The electronic recoil (ER), NR, and surface background are
estimated using the same method as in Ref. [meng2021dark] but with the new
data selections and the ROI cut.

Figure 3:  The $S1$ (left panels) and $S2$ (right panels) spectra in the side-
band (top panels) and ROI (bottom panels) for the 2-hit data, with the data
and corresponding predictions overlaid. the shaded regions represent the
1$\sigma$ uncertainty of the prediction (30%). We also overlay the expected 8B
CE$\nu$NS spectra (scaled up by 50) in the bottom panels, shown in blue solid
lines. The goodness-of-fit p-values of the $S1$ and $S2$ spectra in the side-
band and ROI are all no less than 0.1.

The rate of the AC background is estimated using random $S1$s and $S2$s
identified in the data. The $S2$s are first selected from a waveform
($\sim$1000 per day within 65 to 300 PE), then we search backward for 1.5 ms
for a main $S1$. The 1.5-ms window is choosen so that the corresponding
“activity” cuts are sufficiently similar to those mentioned earlier. The AC
pair is formed when the time difference between the $S1$ and $S2$ is within
[0.9, 1.5] ms, beyond the TPC’s maximum drift time (off-window), to guarantee
that there is no correlation. To enlarge the statistics of the AC samples, a
“scrambled” waveform data set is constructed. The waveform of the selected
$S2$ is concatenated after a 1-ms segment randomly selected from our recorded
data, which on average contains 6.3 (0.01) of the $S1$-like signals with the
$S1$ hit equals to (larger than) 1, primarily from dark noises. This
“scrambled” data get passed to the aforementioned software reconstruction and
data selection. The predicted number of AC events in the ROI in the 2- and
3-hit regions can be found in Table 1. The diffusion cut is the most effective
cut, which suppresses the AC by a factor of 8 or so. The AC model is validated
using the events with the $S2$ in the range from 300 to 800 PE (referred to as
the side-band data) and within the FV, which is dominated by the AC (see Table
2). The comparison between the side-band data and the prediction is given in
Table 2, yielding a good agreement. The comparison between the $S1$ and $S2$
spectra of the prediction and the side-band data for the 2-hit region is shown
in Fig. 3. To be conservative, we take 30%, which is the difference (error-
weighted standard deviation) in the normalized $S2$ spectra, as the systematic
uncertainty of the AC model.

Table 1:  ROI comparison: prediction vs. observation in the optimized $S2$ ranges. Number of the pre- and post-BDT events are listed in separate rows. The observed events after unblinding are shown in the last column.  N${}_{\textrm{hit}}$ |  $S2$ range | BDT | ER | NR | Surf | AC | Total | 8B | Obs
---|---|---|---|---|---|---|---|---|---
[PE] | BKG
2 | 65-230 | pre | 0.04 | 0.10 | 0.14 | 62.43 | 62.71 | 2.32 | 59
post | 0.02 | 0.04 | 0.03 | 1.41 | 1.50 | 1.42 | 1
3 | 65-190 | pre | 0.01 | 0.05 | 0.08 | 0.79 | 0.93 | 0.42 | 2
post | 0.00 | 0.02 | 0.03 | 0.02 | 0.07 | 0.29 | 0
Table 2:  Side-band comparison: prediction vs. observation for $S2$ within [300, 800] PE.  N${}_{\textrm{hit}}$ | Physical | AC | Total | Obs
---|---|---|---|---
1 | 9.4 | 2060.5 | 2069.9 | 2043
2 | 10.1 | 33.8 | 43.9 | 47
3 | 6.9 | 2.2 | 9.1 | 7

A BDT algorithm [hoecker2007tmva] is trained to optimize the 8B CE$\nu$NS
selection against the AC background. The $S2$s of the AC events are mostly
generated out of the fiducial region (such as the surface of electrodes and
the gas region), and the $S1$s are mostly dark noises (see Ref.
[abdukerim2022study]), both having different characters from the physical
events. The input variables of the BDT concern features related to the charge,
width, top-bottom asymmetry, and PMT top patterns of the $S1$ and $S2$
signals. The training and testing samples of the 8B signal in the BDT are from
the WS with the ($S1$, $S2$) distribution following our 8B signal model. The
BDT cut value and the $S2$ range for each $S1$ hit bin are determined by
maximizing the probability of discovering a 8B signal under our background
model, with results summarized in Table 1. The optimized BDT efficiency of the
8B signal is shown in Fig. 1. The BDT reduces the 8B CE$\nu$NS signal (AC
background) by about 39% (98%) and 31% (96%), respectively, for the 2- and
3-hit bins. Most of the rejection power against the AC is gained through the
parameters related to the $S2$ waveform shape and its top charge pattern, and
we observe almost no correlation in the $S1$ and $S2$ discriminants. The
uncertainties of the BDT efficiency to the 8B CE$\nu$NS and the DM signals are
studied using the neutron calibration data. To improve the statistics in the
ROI, especially for $S2$ less than 100 PE, the minor $S2$s of the neutron
double-scatter events are used. A difference of 14% and 13% are observed for
the 2-hit and 3-hit ROI, respectively, taken as the systematic uncertainties.
The systematic uncertainty of the BDT efficiency to the AC background is
estimated by checking the performance on an alternative AC model using a more
traditional approach based on the random pairing of the isolated $S1$s and
$S2$s [abdukerim2022study], leading to an uncertainty of 19% and 18% in the
2-hit and 3-hit bins.

Table 3:  List of the constrained nuisance parameters that are included in the final statistical interpretation (see text), along with the standard deviations of their Gaussian constraints.  Nuisance parameters |  | Stdev. | Estimated by
---|---|---|---
| 2-hit | 3-hit
pre-BDT eff. | $\delta_{\epsilon}$ | 0.14 | WS vs. NR
NR signal rate | $\delta_{s}f_{i}$ | $f_{i}$ |  NEST uncert. [szydagis2022noble]
AC rate | $\delta_{b}$ | 0.30 | Pred. vs side-band
BDT eff. to signal | $\delta_{\textrm{BDT},s}^{i}$ | 0.14 | 0.13 | WS vs. NR
BDT eff. to AC | $\delta_{\textrm{BDT},b}^{i}$ | 0.19 | 0.18 |  Alter. models [abdukerim2022study]
Solar 8B flux | $\delta_{\Phi}$ | 0.04 |  Ref. [aharmim2013combined]
Figure 4:  Top panel: our constraint on the solar neutrino flux using the
CE$\nu$NS analysis, along with the XENON1T results [aprile2021search] using
the same detection channel and the B16-GS98 standard solar model prediction
[vinyoles2017new]. Bottom panel: updated constraints on the DM-nucleon spin-
independent cross section. The red solid and dashed lines represent the
PandaX-4T results from this and the previous searches [meng2021dark],
respectively. The black solid and dashed lines represent the results from
XENON1T with and without optimization in the low-energy region
[aprile2018dark, aprile2021search]. Several results from other experiments
[aprile2019light, akerib2017results] are also shown. The neutrino floors
(probability for an ideal xenon detector to see less-
than-3$\sigma$-significance DM signal) [ruppin2014complementarity] under
different exposure assumptions (1, 10, and 1000 tonne-year from top to bottom)
are shown in the grey shaded regions. The green region represents the
$\pm$1$\sigma$ sensitivity band for the DM search.

The data within the ROI were blinded before we finalized the data selection,
the background and signal models, the ROI, and the BDT optimization. We then
unblinded the data and checked the events before and after applying the BDT.
We show the comparison of the $S1$ and $S2$ spectra between the prediction and
data before applying the BDT in Fig. 3. The observed number of the events in
the ROI for the 2- and 3-hit regions are given in Table 1. After unblinding, 1
(with $S1$=1.6 PE and $S2$=165 PE) and 0 events that survive the BDT are found
in the 2- and 3-hit ROI, respectively.

We perform a simple statistical interpretation based on 2-bin profile
likelihood ratio (PLR) analysis [baxter2021recommended] using the 2- and 3-hit
data. The binned likelihood is defined as 111Penalty term $G(\delta_{\Phi})$
is not used when interpreting results under the signal hypothesis of solar 8B
CE$\nu$NS.:

$\displaystyle\mathcal{L}=$ $\displaystyle
G(\delta_{\epsilon})G(\delta_{s})G(\delta_{b})G(\delta_{\Phi})$ (1)
$\displaystyle\times\left[\prod\limits_{i}G(\delta_{\textrm{BDT},s}^{i})G(\delta_{\textrm{BDT},b}^{i})\frac{\lambda_{i}^{N_{i}}}{N_{i}!}e^{-\lambda_{i}}\right],$

where the index $i$ represents the hit number of $S1$ (2 or 3), and
$\bm{\delta}$ ($\bm{\delta^{i}}$) is series of the constrained nuisance
parameters, which are correlated (independent) between the 2- and 3-hit bins
with a Gaussian penalty $G$ with the mean at zero. The set of parameters
include $\delta_{\epsilon}$, $\delta_{s}$, $\delta_{b}$,
$\delta^{i}_{\textrm{BDT},s}$, $\delta^{i}_{\textrm{BDT},b}$, and
$\delta_{\Phi}$, corresponding to the relative uncertainties of the pre-BDT
efficiency (including the signal reconstruction, data selection, and ROI), the
NR signal rate, the AC background rate, the BDT cut efficiency to the NR
signals, the BDT efficiency to the AC background, and the 8B neutrino flux,
respectively. The 1$\sigma$ values of the nuisance parameters are summarized
in Table 3. The parameter $\delta_{s}$ is factored together with the
fractional uncertainty of the signal rate $f_{i}$ which depends on the the
signal spectrum ($f^{\nu}_{i}$ for the 8B CE$\nu$NS signal and $f^{\chi}_{i}$
for the DM signal), in order to reflect the common origin of $f_{i}$. Typical
numbers of $f_{i}$ are 0.45 (0.60), 0.29 (0.39), and 0.16 (0.24) for 4-GeV/c2
DM, the 8B CE$\nu$NS, and 8-GeV/c2 DM in the 2-hit (3-hit) region.
$\lambda_{i}$ is the expected count while $N_{i}$ is the observed count.
Specifically, under the hypotheses of a) the solar 8B neutrino CE$\nu$NS
without the DM, and b) the low mass DM with the 8B CEvNS background, the
expected counts can be written as:

$\displaystyle\lambda^{\nu}_{i}=$ $\displaystyle
N_{\nu}(1+\delta_{s}f^{\nu}_{i})(1+\delta_{\epsilon})(1+\delta^{i}_{\textrm{BDT},s})+$
(2) $\displaystyle
N_{\textrm{AC}}(1+\delta_{b})(1+\delta_{\epsilon})(1+\delta^{i}_{\textrm{BDT},b})+N_{\textrm{other}},$
$\displaystyle\lambda^{\chi}_{i}=$ $\displaystyle
N_{\chi}(1+\delta_{s}f^{\chi}_{i})(1+\delta_{\epsilon})(1+\delta^{i}_{\textrm{BDT},s})+$
$\displaystyle
N_{\nu}(1+\delta_{s}f^{\nu}_{i})(1+\delta_{\epsilon})(1+\delta^{i}_{\textrm{BDT},s})(1+\delta_{\Phi})+$
$\displaystyle
N_{\textrm{AC}}(1+\delta_{b})(1+\delta_{\epsilon})(1+\delta^{i}_{\textrm{BDT},b})+N_{\textrm{other}},$

where $N_{\nu}$, $N_{\textrm{AC}}$, $N_{\textrm{other}}$, and $N_{\chi}$ are
the nominal numbers of counts for the 8B CE$\nu$NS, AC, other background
events (including ER and neutron), and low mass DM, respectively. The total
backgrounds predicted in the 2- and 3-hit ROI for the solar 8B neutrino search
are 1.50 and 0.07, respectively, in an exposure of 0.48 tonne$\cdot$year, as
shown in Table 1. The observed number of events is consistent with both
background-only hypotheses in searching for the 8B CE$\nu$NS and the low mass
DM in Eqn. 2, representing a probability of 53% and 17% of observing the same
or less number of events than the data, respectively.

Using a similar procedure as in Refs. [meng2021dark, baxter2021recommended],
we give the 90% C.L. upper limit on the solar 8B neutrino flux using the
CE$\nu$NS channel, pushing the upper limit to 9.0$\times$106/cm2/s, in
comparison to (5.46$\pm$ 0.66)$\times$106/cm2/s from the standard solar model
B16-GS98 [vinyoles2017new]. If the signal model adopted by XENON1T
[aprile2021search] is used, the upper limit of the solar 8B neutrino flux will
be lowered by 13%. If the signal model uncertainty ($\delta_{s}$) is
eliminated from the fit, the upper limit will be reduced by 10%. Under the
nominal 8B CE$\nu$NS rate, we also obtain the best constraints on the spin-
independent DM-nucleon cross section with mass in the range of 3 to 9 GeV/c2.
The results are summarized in Fig. 4. In Fig. 4, we also show the 8B neutrino
floor curves from Ref. [ruppin2014complementarity] under ideal background
assumption. The current stage of PandaX has clearly entered into the sensitive
region for neutrinos, so this result could also be cast into interesting
parameter space of neutrino interactions. The lack of CE$\nu$NS excess from
this work and XENON1T [aprile2021search] also motivates further investigations
on the response of LXe TPC to ultralow energy nuclear recoils.

In summary, a search for CE$\nu$NS from the solar 8B neutrinos as well as the
low mass DM-nucleon interactions is performed using the PandaX-4T
commissioning data with 0.48 tonne$\cdot$year exposure. In the analysis, we
have further optimized the data selection and developed various techniques to
lower the energy threshold and to control the accidental background. No
significant excess is observed, leading to the strongest upper limit on the
solar 8B neutrino flux using CE$\nu$NS, and on the spin-independent DM-nucleon
cross section within the mass range from 3 to 9 GeV/c2. This manifests the
potential of PandaX-4T as a highly sensitive multi-purpose dark matter and
astrophysical neutrino observatory.

We would like to thank Matthew Szydagis for useful discussions concerning NEST
model uncertainty. This project is supported in part by grants from National
Science Foundation of China (Nos. 1209061, 12005131, 11905128, 11925502), a
grant from the Ministry of Science and Technology of China (No.
2016YFA0400301), and by Office of Science and Technology, Shanghai Municipal
Government (grant No. 18JC1410200). We thank supports from Double First Class
Plan of the Shanghai Jiao Tong University. We also thank the sponsorship from
the Chinese Academy of Sciences Center for Excellence in Particle Physics
(CCEPP), Hongwen Foundation in Hong Kong, Tencent Foundation in China and
Yangyang Development Fund. Finally, we thank the CJPL administration and the
Yalong River Hydropower Development Company Ltd. for indispensable logistical
support and other help.

Appendix on waveform simulation. To have sufficient high-purity samples for
estimating the efficiency in the ROI and for training the boosted decision
tree (BDT) algorithm, a waveform simulation (WS) is developed, which includes
our best knowledge from the data. The WS not only simulates the $S1$ and $S2$
pulses, but also simulates the accompanying noises that could appear in the
event waveform, such as the PMT afterpulsing, the delayed electrons, the
photo-ionization, and the spurious $S1$s [sorensen2017two,
sorensen2017electron, akerib2020investigation, akerib2021improving].

Figure 5:  The comparisons between the neutron calibration data and WS of $S1$
width (top), $\sigma_{\textrm{pos}}$ (middle) and $S2$ width (bottom)
normalized distributions in the ROI.

The simulation of the $S1$ and $S2$ pulse waveforms is data driven. The
simulated $S1$ pulse waveform is sampled using the real $S1$ hits from the
neutron calibration data with the charge from 20 to 80 PE. The width
distribution of the simulated $S1$s and $S1$s from the neutron calibration
data in the ROI can be found in the top panel of Fig. 5. The simulated $S2$
pulse waveform is re-assembled using the single electron (SE) waveforms
obtained from the data. The SEs are sampled within a circle with a radius
$R_{0}$=40 mm based on their reconstructed positions. $R_{0}$ is tuned to
match the root-mean-square distance of all fired top PMTs, weighted by charge,
from the position of the top PMT that sees the most $S2$ charge
($\sigma_{\textrm{pos}}$). With $R_{0}$=40 mm, the comparison of
$\sigma_{\textrm{pos}}$ between the neutron calibration data and WS are shown
in the middle panel of Fig. 5. The pileup of the SEs is required to follow a
Gaussian distribution with the Gaussian $\sigma$ equals to $\sqrt{2DT}$, where
$D$ is the longitudinal diffusion coefficient in LXe and $T$ is the drift time
of the simulated $S2$. The value of $D$ is obtained to be 28 $\rm{cm}^{2}/s$
by matching the $S2$ width vs. drift time distribution of the neutron
calibration data. The comparison of the $S2$ width distribution between the
neutron calibration data and WS in the ROI can be found in the bottom panel of
Fig. 5.

Figure 6:  Comparison between the $\mathcal{F}_{S2}$ distributions from the WS
(red solid line) and data (black dots with error bars).

Dark counts and noises are included by inserting randomly picked 1-ms-long
waveforms from all the recorded waveforms into the simulated event window. PMT
afterpulsing are already included, since the simulated $S1$ and $S2$ are both
sampled using the waveforms from the data. Delayed electrons and impurity
photo-ionization can cause small delayed $S2$ signals after a large $S2$. The
time profile and probability of such delayed $S2$s are obtained by analyzing
the data waveforms after the main $S2$. The parameter which is mostly
sensitive to the noise and afterglows is the $\mathcal{F}_{\textrm{S2}}$ that
is defined in the main text. Fig. 6 shows the comparison between the
$\mathcal{F}_{S2}$ distributions from the neutron calibration data and the WS.


# Emergent symmetries of relativistic fluid dynamics from local ergodicity

Giorgio Torrieri1 $\phantom{A}{}^{1}$Universidade Estadual de Campinas -
Instituto de Fisica ”Gleb Wataghin”
Rua Sérgio Buarque de Holanda, 777
CEP 13083-859 - Campinas SP
<EMAIL_ADDRESS>

###### Abstract

We show that volume-preserving diffomorphisms and the chemical shift symmetry
defining relativistic lagrangian ideal fluid dynamics can be derived as an
emerging symmetry when ergodicity is assumed to apply locally in a way that is
invariant under smooth spacetime foliations. This can be used as a way to
derive the ideal hydrodynamic limit in a strongly coupled but strongly
fluctuating medium. We comment on the connection with thermalization in small
systems, the Eigenstate thermalization hypothesis and deviations from the
ideal limit.

Relativistic hydrodynamics has proven to be an extremely useful
phenomenological tool kodama ; rom ; heinz  in the study of heavy ion
collisions. This has also reignited the more formal theoretical study of
hydrodynamics denicol ; hydeft , a subject much more complex and subtle on a
conceptual level than one initially thinks. The observation of hydrodynamic
behavior in “small” systems, both in size and number of degrees of freedom
zajc  (but also analogous non-relativistic examples such as ultracold atoms
ultracold  and the “Brazil-nut effect” braznut ) demonstrates that conceptual
ambiguities can still be relevant for resolving mysteries given to us by
experimental data.

One such conceptual issue is that there are several definitions to
hydrodynamics: One usually thinks of it as a coarse-grained, long range
effective theory written in terms of conserved charges and their gradients
kodama ; rom ; denicol . In this approach, hydrodynamic evolution is
explicitly deterministic, and the connection to local thermal equilibrium is
not “built in”, leading to concepts such as hydrodynamic attractors far from
thermal equilibrium attractor ; mrow . The potential ambiguity comes from the
fact that methods from statistical mechanics are necessary to calculate the
equation of state and transport coefficients needed to close the hydrodynamic
system of equations.

Statistical mechanics however is not a deterministic theory, at best
fluctuations vanish in the thermodynamic limit, but in principle it’s only
assumption is the equal probability of microstateskhinchin ; jaynes . One
therefore needs to make sure fluctuations stay “small”, something usually
implicitly assumed in microscopic theories rather than demonstrated ll ; gale
; csernai ; kovfluct ; kapusta ; teaney ; jorgefluct ; zubarev  (Kubo
formulae, for instance, are based on taking limits much larger than the
fluctuation timescale heinz ), though fluctuation-dissipation relations apply
also away from the thermodynamic limit bonanca .

As a perhaps related issue, away from perfect equilibrium the definition of
“equilibrium” becomes itself ambiguous kovtun ; spalinski ; prx ; gavassino ;
bdnk . A full definition of hydrodynamics in this sense, as well as a
connection to non-equilibrium quantum field theory that encodes microscopic
fluctuations and their hydrodynamic response arnold ; gelis ; calzetta , is
still missing.

A still different definition of hydrodynamics is based on a lagrangian
picture, where instead of defining currents in the lab frame we follow every
cell of the fluid as it propagates and deforms. This definition is
particularly elegant in that ideal hydrodynamics is based on local
diffeomorphisms jackiw ; nicolis ; son , making it into a “poor people’s
general relativity” hydeft ; analogue  (to the extent that perhaps the problem
of quantizing gravity can be related, via the enthropic hypothesis jacobson ,
to the problem, described in the previous paragraph, of defining a fluctuating
hydrodynamics where local symmetries still hold zubarev ; deco ).

This work makes a link between this last Lagrangian definition of
hydrodynamics and the previous two, by showing it’s elegant local symmetries
are in fact emerging if one assumes the ergodic hypothesis applies in every
cell of the fluid. This link can then be used to define hydrodynamics in a way
that might be more appropriate to small systems zajc , and as a starting point
to examine the fluctuations and redundancies of bdnk .

Statistical mechanics is based on the emergence of an approximately ergodic
systemll  (or rather khinchin  of the approximation that some observables are
described by such a system), whose evolution is governed by an equation of the
type

$\left.\frac{\Delta x}{\Delta t}\right|_{t-t^{\prime}\simeq\Delta}\simeq\int
P(q^{\prime})dq^{\prime}$ (1)

where $x$ is a microscopic phase space degree of freedom, $t$ is a time
sampled across some average interval $\Delta$,and $q^{\prime}$ is a time-
invariant measure over an ensemble of states.

In other words when sampled under a certain coarse-graining timescale, the
system’s “evolution” can be approximated via a statistical distribution of
microstates. As shown in jaynes ; khinchin , while this is an extremely
fragile assumption, dependent on the connectedness of the phase space
hypersurface constrained by conservation laws, it can be thought of as an
approximate assumption good enough to calculate particular observables (in
particular conserved currents) and their ”typical” state in strongly coupled
systems whose initial conditions are sampled across a broad ensemble.

Thus, for a physical system moving in phase space sufficiently chaotically,
with conservation laws of energy and momentum as well as charge (but not
necessarily particle number), this means

$\left.\frac{\Delta\phi}{\Delta
t}\right|_{t-t^{\prime}=\Delta}\simeq\frac{1}{d\Omega(Q,E)}\sum\delta^{4}_{P^{\mu},P_{macro}^{\mu}(t)}\delta_{Q,Q_{macro}(t)}\delta\left(\sum_{j}^{\infty}p_{j}^{\mu}-P^{\mu}\right)\delta\left(\sum_{j}^{\infty}Q_{j}-Q\right)$
(2)

where $\phi=\\{x_{1,2,....N\rightarrow\infty}\\}$ is some degree of freedom,
$p_{i}$ is the conjugate momentum and $Q_{i}$ some conserved charge and
$d\Omega(E,Q)$ is the normalization for that point in spacetime, ( $\sim
e^{dS}$ where $dS$ is the entropy density of the cell). From this the
equilibrium state in the micro-canonical ensemble. Other ensembles follow when
the number of degrees of freedom and volume are taken to infinity.

It is immediately clear that the ergodic hypothesis is not relativistically
covariant, since there is a “t” coordinate on the left hand side (the right
hand side can be made generally covariant by an on-shell condition). For
“global” equilibrium, this is of course not a problem because that comes with
a frame attached to it.

But this seems in contradiction with the concept of “local” equilibrium, where
every cell of a fluid contains “many” degrees of freedom in ergodic
equilibrium. To an extent, this is not a deep issue since local equilibrium is
not rigorously defined. However, it would be interesting to try to define it
from the ergodic hypothesis and approximate deviations from it. To do so, we
would like to investigate what kind of symmetries are needed to be imposed on
Eq. 2 for a relativistic local ergodic evolution to emerge.

To try to add an element of covariance, we define “t” via a foliation,
$\Sigma_{\mu}$. Different $\Sigma_{\mu}$’s are related via generally non-
inertial transformations. We would like to concentrate on transformations that
respect the causality of the foliation

$d\Sigma_{\mu}=\frac{\partial\Sigma_{\mu}}{\partial\Sigma^{\prime}_{\nu}}d\Sigma^{\prime}_{\mu}\phantom{AA},\phantom{AA}\underbrace{(d\Sigma)^{2}}_{d\tau^{2}},\underbrace{(d\Sigma^{\prime})^{2}}_{d\tau^{\prime
2}}>0\phantom{AA},\phantom{AA}\left.\frac{d}{dt}\right|_{dt\sim\Delta}\rightarrow\left.\frac{d}{d\Sigma_{0}}\right|_{d\tau\sim\Delta}\phantom{AA},\phantom{AA}d\Sigma_{\mu}\det\left(\partial_{\mu}\Sigma_{\nu}\right)^{-1}\leq\Delta$
(3)

A definition of “ideal” local equilibrium is to ensure that Eq. 2 stays true
independently of $\Sigma_{\mu}$ as long as $\Sigma$ is causal. In this
picture, the averaging is done not on $dt\sim\Delta$ but on $d\tau\sim\Delta$,
for each cell. The change in foliation must be smooth enough as to not disturb
the average $\Delta$ (the “microscopic scale”, which in this case is the
maximum allowed radius of curvature of the foliations).

To see if this is possible, let us remember that $p_{\mu}$ and $Q$ are Noether
currents, corresponding to symmetries of the Lagrangian

$p_{\mu}=\int
d^{3}\Sigma^{\nu}T_{\mu\nu}\phantom{AA},\phantom{AA}T_{\mu\nu}=\frac{\partial
L}{\partial{\partial^{\mu}\phi}}\Delta_{\nu}\phi-
g_{\mu\nu}L\phantom{AA},\phantom{AA}\Delta_{\nu}\phi(x_{\mu})=\phi(x_{\mu}+dx_{\nu})$
(4) $Q=\int
d^{3}\Sigma^{\nu}j_{\nu}\phantom{AA},\phantom{AA}j_{\nu}=\frac{\partial
L}{\partial{\partial^{\mu}\phi}}\Delta_{\psi}\phi\phantom{AA},\phantom{AA}\Delta_{\psi}\phi=\phi(x_{\mu})+\delta\phi^{*}(x_{\mu})=|\phi(x)|e^{i\left(\psi(x)+\delta\psi(x)\right)}$
(5)

and $\psi=\\{\psi_{1,2,...N\rightarrow\infty}\\}$ is a complex phase.

These textbook formulae provide us with an intuition of how to model local
equilibrium. All one has to do is to impose ergodicity on every space-time
point and then make this hypothesis foliation-independent. Combining Eq. 2
with 4 and 5 we get

$\left.\frac{\Delta\phi}{\Delta\Sigma_{0}}\right|_{t-t^{\prime}=\Delta}=\frac{1}{d\Omega(dP^{\mu}(\Sigma_{0}),dQ(\Sigma_{0}))}\sum\delta^{4}\left(d\Sigma^{\nu}\underbrace{\frac{\partial
L}{\partial{\partial^{\mu}\phi}}\partial_{\nu}\phi-Ld\Sigma^{\mu}}_{\delta
L(a_{\mu})\sim
a_{\alpha}\partial^{\alpha}\left(\delta_{\nu}^{\mu}L\right)}-dP^{\mu}(\Sigma_{0})\right)\times$
(6) $\ \times\delta\left(d\Sigma^{\mu}\underbrace{\frac{\partial
L}{\partial{\partial^{\mu}\phi}}\frac{\partial\psi}{\partial\Sigma_{0}}}_{\delta
L(\psi)\sim\psi\partial_{\mu}L}-dQ(\Sigma_{0})\right)$

and enforce the covariance of these equations with any change in foliation
that respects causality. Let us therefore use $\delta a_{\mu}$ and
$\delta\psi$ as coarse-graining degrees of freedom for a generic equation of
motion.

$\left.\frac{\Delta\phi}{\Delta\Sigma_{0}}\right|_{t-t^{\prime}=\Delta}=\frac{1}{d\Omega(P^{\mu},Q,\Sigma_{0})}\int
da_{\mu}d\psi\delta^{4}\left(d\Sigma^{\nu}a_{\alpha}\partial^{\alpha}\left(\delta_{\nu}^{\mu}L\right)-dP^{\mu}(\Sigma_{0})\right)\delta\left(d\Sigma^{\mu}\psi\partial_{\mu}L-dQ(\Sigma_{0})\right)$
(7)

Let us now re-write Eq. 7 under a different foliation, and see what kind of
constraints will foliation invariance of the ergodic hypothesis bring.
Provided both foliations are smooth enough (According to Eq. 3) the LHS simply
becomes

$\left.\frac{\Delta\phi}{\Delta
t}\right|_{t-t^{\prime}=\Delta}\rightarrow\frac{d\Sigma^{\prime}_{\nu}}{d\Sigma_{0}}\left.\frac{\Delta\phi}{\Delta\Sigma_{0}}\right|_{\Sigma_{0}-\Sigma_{0}^{\prime}=\Delta}=\frac{\partial\Sigma^{\prime}_{\nu}}{\partial\Sigma_{0}}\left.\frac{\Delta\phi}{\Delta\Sigma_{\nu}}\right|_{x_{\mu}-x_{\mu}^{\prime}<\Delta}$
(8)

now we eliminate the LHS from the system of equations of Eq. 7 in
$\Sigma_{\mu}$ and $\Sigma^{\prime}_{\mu}$. The resulting equation, a ratio of
the RHSs, will be

$\frac{d\Omega(dP_{\mu}^{\prime},dQ^{\prime},\Sigma_{0}^{\prime})}{d\Omega(dP_{\mu},dQ,\Sigma_{0})}=\frac{d\Sigma^{\prime}_{0}}{d\Sigma_{0}}\frac{\int
da_{\mu}d\psi\delta^{4}\left(d\Sigma^{\nu}a_{\alpha}\partial^{\alpha}\left(\delta_{\nu}^{\mu}L\right)-dP^{\mu}(\Sigma_{0})\right)\delta\left(d\Sigma^{\mu}\psi\partial_{\mu}L-dQ(\Sigma_{0})\right)}{\int
da^{\prime}_{\mu}d\psi^{\prime}\delta^{4}\left(d\Sigma^{\prime}_{\nu}a_{\alpha}^{\prime}\partial^{\alpha}\left(\delta_{\nu}^{\mu}L\right)-dP_{\mu}^{\prime}(\Sigma_{0}^{\prime})\right)\delta\left(d\Sigma_{\mu}^{\prime}\psi^{\prime}\partial^{\mu}L-dQ^{\prime}(\Sigma_{0}^{\prime})\right)}$
(9)

One can then use the classic $\delta-$function relation We also take advantage
of the fact that $\Sigma_{\mu}$ are causal, an parametrize $d\Sigma_{\mu}$ by
three Cartesian coordinates $\Phi_{I=1,2,3}$ such that the tangent vector

$d\Sigma_{\mu}=\epsilon_{\mu\nu\alpha\beta}\frac{\partial\Sigma^{\nu}}{\partial\Phi_{1}}\frac{\partial\Sigma^{\alpha}}{\partial\Phi_{2}}\frac{\partial\Sigma^{\beta}}{\partial\Phi_{3}}d\Phi_{1}d\Phi_{2}d\Phi_{3}$
(10)

where the latin letters are 1,2,3 and 0 would be the time coordinate is a
time-like 3-volume element for a certain normalization. Arbitrary changes in
foliation can also be decomposed in

$\frac{\partial\Sigma^{\prime}_{\mu}}{\partial\Sigma_{\nu}}=\Lambda_{\mu}^{\nu}\det\frac{d\Phi^{\prime}_{I}}{d\Phi_{J}}\phantom{AA},\phantom{AA}\det\Lambda_{\mu}^{\nu}=1$
(11)

where $\Lambda_{\mu}^{\nu}$ is a Lorentz transformation and
$\det\frac{d\Phi_{I}}{d\Phi_{J}}$ a rescaling of the volume. Physically,
$\Lambda_{\mu}^{\nu}$ moves between the frame
$d\Sigma_{rest}^{\mu}=d\Phi_{1}d\Phi_{2}d\Phi_{3}(1,\vec{0})$.

We can now try define an effective Lagrangian that “automatically” solves Eq.
9 at every time step, thereby guaranteeing Eq. 6 holds in every frame. Good
coarse-graining variables are generators of the transformation

$L(\phi)\simeq
L_{eff}\left(a_{\mu}(\Sigma),\psi(\Sigma)\right)\phantom{AA},\phantom{AA}L_{eff}\left(a_{\mu}(\Sigma),\psi(\Sigma)\right)=L_{eff}\left(a_{\mu}(\Sigma^{\prime}),\psi(\Sigma^{\prime})\right)$
(12)

With the latter equality being enforced by solving Eq. 9.

Concentrating on the momentum current conservation, we notice that the
dependence on foliation changes of the ratio of $\Omega(...,P_{\mu})$ is the
same as $\delta(...,P_{\mu})$, and the shift in the Lagrangian can be reduced
to a local symmetry shift.

The only change possible therefore is that of the foliation inside the
$\delta$-function $\delta^{4}(d^{3}\Sigma_{\mu},...)$. One can use the classic
relations of $\delta-$functions

$\delta((f(x_{i})))=\sum_{i}\underbrace{\frac{\delta(x_{i}-a_{i})}{f^{\prime}(x_{i}=a_{i})}}_{f(a_{i})=0}\phantom{AA},\phantom{AA}\phi_{I}^{\prime}=\frac{\partial_{\alpha}\Sigma^{\prime}_{I}}{\partial^{\alpha}\Sigma^{J}}\Phi_{J}\phantom{AA},\phantom{AA}\delta^{4}(\Sigma_{\mu})=\mathrm{det}\left|\frac{\partial\Sigma^{\mu}}{\partial\Sigma^{\nu}}\right|\delta^{4}(\Sigma^{\prime}_{\mu})$
(13)

to see that the ratio of
$\delta^{4}\left(d\Sigma^{\nu}a_{\alpha}\partial^{\alpha}\left(\delta_{\nu}^{\mu}L\right)-dP^{\mu}(\Sigma_{0})\right)$
in Eq. 9 will reduce to a ratio of
$\det\frac{\partial\Phi_{I}}{\partial\Phi_{J}}$

  * •

Assumes $d\Omega$ is differentiable across changes between $\Sigma$ and
$\Sigma^{\prime}$ so
$d\Omega(P^{\mu},Q,\Sigma^{\prime}_{0})/d\Omega(P^{\mu},Q,\Sigma_{0})\simeq
d\Sigma^{\prime}_{0}/d\Sigma_{0}$,$Q$ transforms as a scalar and $P_{\mu}$ as
a 4-vector under changes in foliation

  * •

Assumes the effective Lagrangian $L_{eff}$ is invariant under transformations
leaving $\det\frac{\partial\Phi_{I}}{\partial\Phi_{J}}$ unchanged

$L(a_{\mu},\psi)\rightarrow
L_{eff}(\phi_{I},\psi)\phantom{AA},\phantom{AA}L(\phi_{I},\psi)=L_{eff}\left(\underbrace{\frac{\partial{\phi^{\prime}_{I}}}{\partial\phi_{J}}}_{\det\frac{\partial{\phi^{\prime}_{I}}}{\partial\phi_{J}}=1}\phi_{J},\psi\right)$
(14)

  * •

Introduces a rescaling, allowed by the above properties of $d\Omega$ and
$L_{eff}$

$\Phi_{I}\rightarrow
T(x_{\mu})\phi_{I}\phantom{AA},\phantom{AA}d\Sigma=\exp\det_{IJ}\underbrace{\partial_{\mu}\phi_{I}\partial^{\mu}\phi_{J}}_{\equiv
b}$ (15)

where $b$ is defined as in son . $T$ is a scale of units of energy, identified
melag ; hydeft  with a “microscopic” temperature, dependent on the microscopic
degeneracy $g$, $T=(2g)^{-1}b\frac{dL_{eff}}{db}$.

all $d\Sigma_{0}/d\Sigma_{0}^{\prime}$ and ratios of
$\delta^{4}\left(d\Sigma^{\nu}a_{\alpha}\partial^{\alpha}\left(\delta_{\nu}^{\mu}L\right)-dP^{\mu}(\Sigma_{0})\right)$
cancel out and 9 reduces automatically to

$1=\frac{\delta\left(d\Sigma^{\mu}\psi\partial_{\mu}L-dQ(\Sigma_{0})\right)}{\delta\left(d\Sigma_{\mu}^{\prime}\psi^{\prime}\partial^{\mu}L-dQ^{\prime}(\Sigma_{0}^{\prime})\right)}$
(16)

To construct the symmetries allowing Eq. 16 to be satisfied automatically we
remember that since $d\Sigma^{\mu}\partial_{\mu}$ is a scalar, we can evaluate
it in the frame where
$d\Sigma^{\mu}=d\Phi_{1}d\Phi_{2}d\Phi_{3}\left(1,\vec{0}\right)$, where it
becomes clear that $\partial_{\mu}L$ will reduce to gradients of
$\partial_{\mu}\Phi_{1,2,3}$ and $\partial_{\mu}\Psi$. Since these are
arbitrary transformations, imposing Eq. 16 is of course equivalent to imposing
invariance of $\psi$ under arbitrary shifts of functions of $\Phi_{I}$.

Summarizing this main result of our work, for the local ergodicity condition
Eq. 7 to apply for a generic foliation $\Sigma$, the effective Lagrangian
$L_{eff}$, written in terms of generators of local space translations
$\phi_{I}$ and internal symmetry phases $\psi$, must be invariant under any
$\phi_{I}\rightarrow\phi^{\prime}_{I}$ leaving
$\det_{IJ}\frac{\partial\phi^{\prime}_{I}}{\partial\phi_{J}}$ and
$\psi\rightarrow\psi+f(\phi_{I})$ invariant. These of coarse are exactly the
symmetries in jackiw ; nicolis ; son .

The identification of conserved currents in terms of $L_{eff}$ was done in son
by the guaranteed existence, under the above set of diffeomorphisms of the
killing 4-vector defined by

$\
u_{\mu}u^{\mu}=-1\phantom{AA},\phantom{AA}u_{\mu}\partial^{\mu}\Phi_{I=1,2,3}=0$

which defines the boost from a generic $d\Sigma_{\mu}$ to one where it is of
the form $d\phi_{1}d\phi_{2}d\phi_{3}(1,\vec{0})$. by inspection

$\partial_{\mu}s^{\mu}=\partial_{\mu}J^{\mu}=0\phantom{AA},\phantom{AA}s^{\mu}=bu^{\mu}\phantom{AA},\phantom{AA}J^{\mu}=dQu^{\mu}$
(17)

While this ideal hydrodynamics is the same as the usual one kodama ; rom ;
denicol , this way of arriving at the ideal hydrodynamics as a limit is
profoundly different from the usual approach of coarse-graining Boltzmann-type
equations around conservation laws, one perhaps more useful to study
hydrodynamics in small systems such as droplets of quark-gluon plasma, zajc
ultracold atoms ultracold  and perhaps the everyday phenomenon of “the
Brazilian nut effect” braznut .

In the usual approach denicol  hydrodynamics is a limit of a deterministic
transport equation valid in the ensemble average limit. This means a hierarchy

$L_{micro}\ll\underbrace{L_{diss}}_{\sim\eta/(sT)}\ll\underbrace{L_{macro}}_{\sim(\partial
u_{\mu})^{-1}}$ (18)

where $L_{micro}$, the origin of stochasticity is neglected and the effective
theory is obtained by a coarse-graining around $L_{diss}/L_{macro}$
($L_{diss}$ determines mean free path, or viscosity to entropy ratio
$\eta/s$).

In our approach, we start with the ergodic hypothesis to hold approximately in
every cell at a scale $\Delta$ , but we do not consider fluctuations to be
necessarily small; In fact, the deterministic continuum is coincident with the
onset of the regression to the mean by the law of large numbers. Such “coarse
graining around ergodicity” in the ideal limit means

$L_{diss}=0\phantom{AA},\phantom{AA}L_{micro}\equiv\Delta\ll L_{macro}$ (19)

If one could move beyond $L_{diss}=0$ this could lead To a limit to
hydrodynamics that depends very differently on the thermal and mean free path
scale. The main point here is that

(i)

a strongly coupled system could be ”close the ergodic limit” at every cell
even if the number of dofs in each cell is small.

(ii)

Since ergodicity and frame Independence are different concepts, in such a
limit, foliation independence should still be valid. It is in fact far from
clear that, provided the system is strongly coupled and chaotic, decreasing
the number of DoFs decreases the applicability of the ergodic hypothesis. In
this sense, fluctuations should respect foliation independence. Approximate
foliation independence means fluctuations are inherently non-linear and in
fact often help the system thermalize. This is particularly appealing if
fluctuation-dissipation holds bonanca .

To expand on point (i), the considerations above are particularly relevant
when the system is small enough for quantum effects to be relevant. In fact,
One can use the direct connection between the Eigenstate thermalization
hypothesis eth  (or similarly “Berry’s conjecture, see beceth  and references
therein) and the microcanonical density matrix to generalize the derivation
here to a quantum system. A generic mixed density matrix commuting with the
Hamiltonian of a theory with a conserved charge and energy-momentum is
characterized by the density matrix

$\hat{\rho}=\delta_{P,P^{\prime}}\delta_{Q,Q^{\prime}}c_{P,Q}c_{P^{\prime},Q^{\prime}}^{*}|i,P_{\mu},Q><j,P_{\mu},Q|\phantom{AA},\phantom{AA}\hat{U}^{-1}\hat{\rho}\hat{U}=\hat{\rho}$
(20)

where

$\hat{U}=\exp\left[i\hat{P}^{\nu}\delta{x}_{\nu}\right]\exp\left[i\theta\delta\hat{Q}\right]|$
(21)

the phase invariance $\theta$ in a first quantized theory just reflects
unitarity and $J_{\mu\nu}$ are simply generators of momentum and angular
momentum. For a highly mixed thermalized system, $\rho$ becomes a pseudo-
random matrix, averaged to $c_{P,Q}c_{P^{\prime},Q^{\prime}}^{*}\simeq
1/d\Sigma(P,Q)$, making it indistinguishable from the microcanonical ensemble
examined in this paper.

Applying the Eigenstate thermalization hypothesis to every cell in every
foliation is equivalent to promoting $J_{\mu\nu},\theta,P,Q$ to functions of
$x_{\mu}$ and imposing foliation independence on the “pseudo-randomness” of
$\hat{\rho}$.

$\left.\frac{d\hat{\rho}}{d\Sigma_{0}}\right|_{\Sigma_{0}-\Sigma_{0}^{\prime}\simeq\Delta}=0\phantom{AA},\phantom{AA}\hat{\rho}\simeq\frac{1}{d\Sigma}\hat{\delta}_{E,E^{\prime}}\hat{\delta}_{Q,Q^{\prime}}$
(22)

and also

$\hat{U}^{-1}(x)\hat{\rho}\hat{U}(x)\simeq\hat{\rho}\phantom{AA},\phantom{AA}\hat{U}(x)=\exp\left[i\hat{T}^{\mu\nu}d^{3}\Sigma_{\mu}\delta{x}_{\nu}\right]\exp\left[i\partial_{\alpha}\theta
d^{3}\Sigma^{\alpha}\delta\hat{Q}\right]$ (23)

for arbitrary $d^{3}\Sigma_{\mu}$. By imposing $\Sigma$-invariance on the
first equation it is not too difficult to see that an equation equivalent to
Eq. 9 can be derived, and hence quantum dynamics will be equivalent to the
classical microcanonical one.

What this reasoning suggests is that if the system can be divided into cells
where the Eigenstate thermalization hypothesis applies, then a dynamics
equivalent to ideal fluid dynamics will naturally emerge even if the number of
degrees of freedom is comparatively low and fluctuations are big. In the
context of heavy ion physics, therefore, the applicability of the Eigenstate
thermalization hypothesis suggested in beceth  should go hand in hand in the
onset of hydrodynamic behavior where spacial anisotropy is also present.
Whether the Eigenstate thermalization hypothesis can be applied to QCD in the
strongly coupled regime of course remains to be understood quasi .

Regarding point (ii), to try to relax the $L_{diss}=0$ limit of Eq. 19, let us
now imagine the system is approximately locally ergodic, defined by the idea
that deviations from ergodicity are within an standard fluctuation. $\phi_{I}$
can then be thought of not as coordinates but as probability distributions,
each characterized by a killing vector $u_{\mu}$. One heuristically arrive at
the picture in bdnk . To do so one can use escort distributions on top of the
ergodic distribution escort

$\phi_{I}\rightarrow\hat{\phi}_{I}\equiv\exp\left[\phi_{I}-\frac{d\ln\mathcal{Z}}{d\Sigma_{I}}\right]\phantom{AA},\phantom{AA}Z=\mathrm{Tr}\exp\left[d\Sigma_{\mu}\beta_{\nu}\hat{T}^{\mu\nu}\right]$
(24)

One can then link lagrangian hydrodynamics to Zubarev relativistic
hydrodynamics as developed in zubecc .

In addition let us suppose in accordance to (ii) that the fluctuation-
dissipation theorem still applies, so fluctuations around equilibrium give a
good estimate of deviations from ergodicity. Supposing fluctuation-dissipation
applies is equivalent to supposing that while $b$ as defined in Eq. 17 is not
conserved, the probability of a cell evolving from a starting configuration at
$b=b_{0},\tau=\tau_{0}$ to $b=b_{0}+db$ at
$\tau=d\tau=d\Sigma_{\mu}d\Sigma^{\mu}$ is

$\frac{P\left(\phi_{I}=\phi_{I0}+d\phi_{I},\tau=\tau_{0}+d\tau|\phi_{I}=\phi_{I0},\tau=\tau_{0}\right)}{P\left(\phi_{I}=\phi_{I0},\tau=\tau_{0}+d\tau|\phi_{I}=\phi_{I0}+d\phi_{I},\tau=\tau_{0}\right)}=\frac{f(b+db)}{f(b)}$
(25)

i.e. a function of the entropy only. one can easily check that, while this
dynamics need not be compatible with Eq. 6, it remains invariant under
foliation changes. Keeping the idea that the number of miscrostates is
$\sim\exp(b)$ forces $f(b)$ to be an exponential as well. It is in fact the
dynamics of zubarev  expressed in Lagrangian hydrodynamics. The multiple
definitions of $\phi_{I}$ should therefore accommodate the gauge-like symmetry
of zubarev  described in bdnk . Here, it is worth pointing out that if one
uses local ergodicity to describe microscopic fluctuations, it becomes obvious
that, as pointed out in bdnk , only fluctuations of the total $T_{\mu\nu}$ are
physical, while fluctuations of $u_{\mu},e,p,\Pi_{\mu\nu}$ on their own are
essentially equivalent to constrained gauge redundancies, ensembles cannot
depend on them individually. In the lagrangian picture this is evident from
the fact that nicolis ; son  all perturbative degrees of freedom are Goldstone
modes of the symmetries emergent from ergodicity. Distinguishing a non-
hydrodynamic excitation of $\Pi_{\mu\nu}$ from a fluctuation in equilibrium
$u_{\mu},e$ is only possible when the the ergodic limit is exact in every cell
as well as in the ensemble average, i.e. when the number of microscopic DoFs
is infinite in every coarse-grained cell.

It is also worth pointing out that any long-range potential between degrees of
freedom will, by the fluctuation dissipation theorem applied to the
microcanonical ensemble bonanca  be indistinguishable from an anisotropic
collective flow in it’s sampling dynamics. This leads exactly to the picture
advocated around eq. (36) bdnk , where a large class of deformations of the
phase space function has the potential to make Vlasov-type and Boltzmann-type
terms in kinetic theory cancel out, leading to an ensemble of equivalent local
equilibrium states. Superficially this looks like the plasma instability
scenario mrow , but the “equivalent equilibrium” states are really
indistinguishable from bona fide equilibrium. One can see this by considering
that if local ergodicity is approximately respected in each cell as well as
across ensemble averages, hydrodynamic signatures should be expected to
translate across fluctuations,is their fluctuation should reflect initial
state fluctuations, as indeed seems to be the case kodama ; vogel . If
hydrodynamical observables show thermalization driven by plasma instabilities,
one would expect parametrically larger fluctuations shur .

Making a link of this work with hydrodynamics with spin lisa ; hydeft  is a
non-trivial endeavor, because of the fact that the interplay of spin with
angular momentum lead to non-local phase space hypersurfaces with a high
degree of non-trivial connectedness khinchin . Nevertheless, it is possible
that hydrodynamics with spin can only be defined this way, given the fact that
spin-vorticity coupling alters microstate distributions at the same order as
hydrodynamic fluctuations hydeft , something arising as non-locality from
transport models nora .

Finally, as a wild speculation taking inspiration from jacobson , the addition
of a horizon term to the entropy, $d\Omega$, and a generalization of the
ergodicity hypothesis from time averages to worldline/proper time averages
might open the way to an enthropic gravity scenario where the equivalence
principle is respected exactly, as suggested in zubarev ; deco .

In conclusion we have shown that the local symmetries associated with ideal
lagrangian hydrodynamics can be thought of as emerging from a hypothesis of
local ergodicity strong enough to be invariant under smooth spacetime
foliations. This opens a way of thinking about hydrodynamics which is
explicitly ”non-perturbative” with respect to fluctuations and more
appropriate to study the onset of hydrodynamic behavior in small systems.

Acknowledgements GT thanks CNPQ bolsa de produtividade 306152/2020-7, bolsa
FAPESP 2021/01700-2 and participation in tematico FAPESP, 2017/05685-2. I also
thank Dam Thanh Son for providing a key insight that led to this development
during his visit to Unicamp.

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